1-4. a: 40 b: 6 c: 7 d: a: 3 b: 5 c: 6 d: 2

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1 Lesson Shapes (a), (c), (d), and (e) are rectangles a: 40 b: 6 c: 7 d: a: 3 b: 5 c: 6 d: a: 22a + 28 b:!23x!17 c: x 2 + 5x d: x 2 + 8x 1-7. Possibilities: Goes to bank, gets money from parent, gets paid; buys lunch, goes shopping, pays a bill, Lesson Answers vary. Possible responses include How many sides does it have?, Does it have a right angle?, Are any sides parallel? Answers vary. Possible responses include They have 3 sides of equal length and They have 3 angles of equal measure a: 3 b: 2 c: a: x = 7 b: c = 4.5 c: x = 16 d: k = a: 12 b: 35 c: 24 d: 7 Lesson c is correct; x = No if the points are collinear then they will not form a triangle See answers in bold below. y = x! 3 x y a: 55.5 sq. units b: 42 sq. units a: 5 b: 12 c: 7 Selected Answers 1

2 Lesson a: x = = b: No solution c: x 6.44 d: x = Yes, his plants will be dead. If his plants are indoors, they will be dead because he will be gone for 2 weeks and so he did not water them at least once a week. If he left them outdoors, they will still be dead because it has not rained for 2 weeks, so he needed to water them once a week as well a: y = 2 3 x! 4 b: y =! a: 6x + 6 b: 6x + 6 = 78, so x = 12 and the rectangle is 15 cm by 24 cm. ( )( 2 + 3) = 360 c: 2! a:! 5 3 b: 6 3 = 2 1 = 2 c:! 6 3 =! 1 2 d: 0 7 = 0 Lesson a: 100 b: 170 c: a: Yes, it is correct because the two angles make up a 90 angle. b: x = 33! so one angle is = 23 while the other is 2(33) + 1 = Perimeter: 74 centimeters, Area: 231 cm a: y = 5 b: r = 12 c: a = 6 d: m = While there are an infinite number of rectangles, possible dimensions with integral measurements are: 1 by 24 (perimeter = 50 units), 2 by 12 (perimeter = 28 units), 3 by 8 (perimeter = 22 units), and 4 by 6 (perimeter = 20 units). 2 Core Connections Geometry

3 Lesson See graph at right. y y = 3x! a: 27 48! 56.3% b: = 13 1 c: 0 d: 5 8! 56% 4 4 y =! 2 3 x x x! 2 + 2x + 6 = 67, x = 9, so 5(9) 2 = 43 miles 4 y =!4x a: x = 3.75 b: x = 3 c: 24 d: x = 3 e: x f: x = The flag would need to be a rectangle. The height of the cylinder would match the height of the rectangle along the pole, and the cylinder s radius would match the width of the rectangle. Lesson Yes; yes; no a: Reflection b: Translation (or two reflections over parallel lines) c: Rotation or rotation and translation d: Rotation or rotation and translation depending on point of rotation e: Reflection f: Reflection and then translation or rotation or both x +10x + 3 = 52, so x = 2. Side lengths are 19, 10, and a: Area 16 square units b: Area 15 square units a: y = 4 b: y = 25 c: y = 2 Selected Answers 3

4 Lesson a: Yes b: 2 5 c: 3 5 d: 1 5 e: a: y = 5 2 x! 3 b: y = 3 2 x C a: p =!5 b: w = 20 c: x = 14 d: y = See answers in bold below. y = 3x + 2 x y Lesson a: Yes. It has four sides. mab = mcd = 1 2 and mbc = mad =!2, so each pair of consecutive sides is perpendicular and forms 90 angles. b: A (4, 3), B (6, 1), C ( 2, 5), D ( 4, 1) a: x =!4.75 b: x =!94 c: x! 1.14 d: a = a: There are 10 combinations: a & b, a & c, a & d, a & e, b & c, b & d, b & e, c & d, c & e, d & e b: Yes. If the outcomes are equally likely, we can use the theoretical probability computation in the Math Notes box in Lesson c: 3 10 d: 9 10 e: The outcomes that satisfy part (c) includes the outcomes that satisfy (b), but there are others on the (c) list as well a: y = 4 3 x! b: The resulting line coincides with the original line; y = 4 3 x! 2 c: The image is parallel; y = 4 3 x! 7 d: They are parallel, because they all have a slope of 4 3. e: y =! 3 4 x Core Connections Geometry

5 Lesson ! ; No, this probability is very small a: (9, 3) b: (3, 3) c: ( 2, 7) d: ( 52, 1483) a: 10 square units b: 20 square units c: 208,680 square units a: b: A B P l c: d: C D Q m a: The orientation of the hexagon does not change. b: The orientation of the hexagon does not change. c: There are 6 lines of symmetry, through opposite vertices and through the midpoints of opposite sides. Selected Answers 5

6 Lesson (a) and (b) are perpendicular, while (b) and (c) are parallel a: One possibility: 4(5x + 2) = 48 b: x = 2 c: 12!12 = 144 square units a: 4 52 = 1 13 b: = 1 4 c: 1 52 d: = a: It looks the same as the original. b: Solution should be any value of 45k where k is an integer. c: circle a: b: A l B P c: d: Q C D m 6 Core Connections Geometry

7 Lesson Carol: only inside circle #2, Bob: outside both circles, Pedro: only inside circle #1. In order to belong to the intersection of both circles, a person would need to have long hair and study a lot for class a: Sandy s probability = 2 4, while Robert s is 5 3. Therefore, Robert has a greater chance. b: Sandy (Sandy s probability = 1 while Robert s is 0) c: Sandy s probability = 4 3, while Robert s is 5 3. Therefore, Sandy is more likely to select a shape with two sides that are parallel a: x =! 33 9 =! 11 3 b: x = 5 and x =! 2 3 c: x = 1 d: x = a: heart b: square c: hexagon d: Answer vary a: ( 6, 3) b: The vertices are: (6, 2), (2, 3), (5, 6) c: (8, 3) Lesson Isosceles: (a) and (c); Scalene: (b) a: 7 16 c: % b: % 6% d: 6 16 = % a: isosceles triangle b: pentagon c: parallelogram d: obtuse scalene triangle e: isosceles right triangle f: trapezoid Reflection only: A, B, C, D, E, M, T, U, V, W, Y Rotation only: N, S, Z Intersection only: H, I, O, X Outside both regions: F, G, J, K, L, P, Q, R a: x =!2 b: x = 2 3 = c: x = 3 d: No solution Selected Answers 7

8 Lesson a: 33 sq. cm b: 33x sq. units c: 33x x + 8 sq. units 2-9. a: 1 2 b: 2 6, parallelogram and square a: Isosceles triangle b: Equilateral triangle c: Parallelogram KITE ISOSCELES TRIANGLE REGULAR HEXAGON RHOMBUS Answers vary. The left circle could be equilateral, and the right could be quadrilateral. Assuming this, you could add an equilateral hexagon to the left, a rhombus to the intersection, and a rectangle to the right circle. Lesson a: Vertical angles, equal measure, 3x + 5 = 5x! 57, x = 31º b: Straight angle pair, supplementary, 2x + 4x +150 = 180, x = 5º a: m!b = m!c because the line of symmetry must pass through A (according to the marked sides of equal length) and these angles are on opposite sides of the line of symmetry. b: Since they are equal, m!b = 1 2 (124 ) = 62. c: 71 + x = 180, x = a: Square b: ( 4, 5), (1, 5), ( 4, 0), (1, 0) y = x!1; No, because 1! 3" a: Vertical; they have equal measure. b: They form a Z. 8 Core Connections Geometry

9 Lesson a: ( 2, 3) b: ( 2, 3); yes a: 20 square units b: 2,600 square units; subtract the x- and y-coordinates to find the length of the two sides a: We do not know the angles measures are equal, because we do not know if! BD ##" $ EG!##". b: The diagram does not have parallel line marks a: x = 17.5 (corresponding angles) b: x = 5 (multiple relationships possible) a: 12 boys b: 22 girls c: 2 3 d: 7 boys left, 23 students, so a: an isosceles triangle b: a rectangle Lesson a: b: The slopes are 1 2 and! 2 3. Since the slopes are not opposite reciprocals, the lines must not be perpendicular (3, 1), (7, 1) They used different units The lines are parallel, so they do not intersect. Therefore, there is no solution. Selected Answers 9

10 Lesson x = 7º a: x = 10 units b: x = 6 c: x = 20! d: x = 10! a: x = 4 and y = 18 b: x =!13 and y = a: It should be a triangle with horizontal base of length 4 and vertical base of length 3. b:! 4 3 c: Any equation of the form y =! 4 3 x + b Lesson The acute and isosceles triangles Reasoning will vary. a = 118, b = 118, c = 32, d = a: 15 b: x = 12, m!d = 4(12 ) + 2 = 50 c: It is equilateral a: A!("6, "3), B!("2, "1), and C!("5, "7) b: B!!(8, 13) c: A!!! (3, " 6) a: Yes, because the slopes are opposite reciprocals. b: y = 1 2 x + 5 c: Any equation of the form y =!2x + b for all real b-values. 10 Core Connections Geometry

11 Lesson a: 8x 2! 26x! 7 b: 10x x!14 c: 4x 2! 47x + 33 d:!6x 2 +17x! area = 28 square feet a: x = 8, right angle is 90 b: x = 20, straight angle is 180 c: x = 20, sum of angles is 180 d: x = 60, sum of angles is Daniel is correct because the definition of a rectangle is a quadrilateral with four right angles. Since a square has four sides and four right angles, it must be a rectangle a: = 62.9% b: = 42? ; she needs to attempt about 67 pancakes. c: She should add three banana pancakes to make the probability of banana Lesson No, it would take 10 months for Sarita to catch up to Berti The unshaded triangle is half the area of the rectangle (0.5(8)(17) = 68 sq. in.), so the shaded area is the other half a: Because when you are not standing up straight, you have changed your height, and you will not get a true measure of your height. b: Diagram (1) is correct. c: No, you measure from the very bottom to the very top a: If it rains, then Mr. Spelling is unhappy. b: If you add two even numbers together, then the result is even. c: If it is Tuesday, then Marla has a piano lesson a: 8x +13 b: 2x + 3 c: 3x 2! 5x!12 c: 13x x Selected Answers 11

12 Lesson a: 7 2 = 49 sq cm b: 0.5(10)4 = 20 sq. in. c: 0.5(16 + 8)6 = 72 sq ft a: 15x x b: x 2 + 5x + 6 c: 3x 2! x!10 c: 10x 2! 3x! See graph at right. ( 3, 0) and (0, 3) a: Isosceles Trapezoid because two sides are parallel and the other two sides are the same length. b: A!(7, "2), B!(8, "4), C!(2, "4), D!(3, "2) c: 10 square units a: = 3 13 b: = 5 13 c: 2 52 = 1 26 d: 0 Lesson units a: (1): (5, 3), (2): (2, 6) b: p: y = 2x + 8 ; q: y =! 1 2 x + 3 c: The slopes indicate that the lines are perpendicular. d: The solution should be ( 2, 4) a: Right triangle; slopes are opposite reciprocals. b: 20 square units c: 23.4 units See diagram at right See graph at right. a: It is a trapezoid because it has exactly one pair of parallel sides. b: A!("2,!"1),! B!("5,!0), C!("5,!2), D!("2,!6) c: A!!(1,!2) and C!!("2,!5) d: 1 2 (3)(2 + 7) = 13.5 units y x y x 12 Core Connections Geometry

13 Lesson a: ( 2, 5) b: (1, 5) c: ( 12, 14) d: (2, 2) a: 7 8 b: 3 8 c: Height = 12 feet; Using the Pythagorean Theorem, area = 1 2 (12)( ) = 210 sq. feet a: x = 28.5, Triangle Angle Sum Theorem b: x = 23, relationships used varies c: x = 68, corresponding angles are congruent because the lines are parallel and base angles of an isosceles triangle are congruent " and 21" Lesson a: D is not similar. AB = 5, BC = 4, AC = 3 b: A! B! = 100 = 10 units, B! C! = 8 units, and A! C! = 6 units. c: A = 24 sq. units; P = 24 units 3-6. a: x = 18 b: x = 3 c: x = 6 d: x = a: 30, 40, 110 b: Obtuse scalene triangle 3-8. a: 4 5, y = 4 5 x b: MU = 41! 6.40 units c: One is a ratio (slope) while the other is a length (distance) a: triangle inequality b: Pythagorean Theorem c: base angles not equal a: If a shape is an equilateral triangle, then it has 120 rotation symmetry. b: If a shape is a rectangle, then the shape is a parallelogram. c: If a shape is a trapezoid, then the area of the shape is half the sum of its bases multiplied by its height. Selected Answers 13

14 Lesson Result should be 12 units tall and 16 units wide a: The 15 corresponds to the 6, while the 20 corresponds to the 8. Multiple equivalent ratios are possible. One possibility: 15 6 = 20 8 = 2.5 b: 25 and 10; = 2.5 ; yes Yes they are parallel because they have the same slope:! a: 6x 2! 8x b: 2x 2 + x!15 c: 4x 2! 25 d: 2x 3! 5x 2! 3x x = 10, y = No, this is not convincing. While the facts are each correct, the conclusion is not based on the facts. As stated in Fact #2, a square is a rectangle because it has four right angles. However, a rhombus does not have to have four right angles, so therefore there is not enough evidence that a rhombus is a rectangle. Lesson a: Zoom factor: 0.5; The sides are only half as long, so the side corresponding to the 16 must become 8, and the side corresponding to the 11 must become 5.5. b: It is 1:1 because it is congruent P(original) = 18 units and P(new) = 36 units; A(original) = 18 sq. units and A(new) = 72 sq. units. The enlarged perimeter is 2 times greater. The enlarged area is not 2 times greater. Notice that the enlarged area is 4 times greater a: x = 42 5 = 8.4 b: m = 22 c: t = 12.5 d: x = 3 2 = a: y = 3! 3 5 x b: A = 7.5 sq. units; P = ! 13.8 c: y = x a: If the lines have the same slope, then they are parallel. b: If a line is vertical, then the slope is undefined. c: If lines have slopes 2 3 and! 2 3, then they are perpendicular a: alt. int. angles b: vertical angles c: corresponding angles d: straight angle (or supplementary) 14 Core Connections Geometry

15 Lesson a: f = 9 b: g = 18 c: h = a: 180! 38! 63 = 79 and 180! 38! 79 = 63 ; corresponding angles are equal. b: Upon inspection, all unmarked angles are the same since the difference with 180 will be the same a: Frank: 0.25x = y ; Alice: 0.40x +1.5 = y b: They will be 3 years old a: If a rectangle has base x and height 2x, then the area is 2x 2. b: If a rectangle has base x and height 3y, then the perimeter is 2x + 6y. c: If a rectangle has base of 2 feet and a height of 3 feet, then the area is 864 sq. inches In theory, 3 < x < 13 but some of these lengths are not practical a: The coordinates of the image are A( 6, 4), B(10, 4), C(10, 6), D(2, 12), E( 6, 6) b: Perimeters = 28 and 56 units; areas = 52 and 208 sq. units Selected Answers 15

16 Lesson a: Yes, since all trees are green and the oak is a tree. b: No, only trees must be green according to the statement. c: No, the second statement reverses the first a: Yes, AA ~. Dilate from right vertex. b: Yes, AA ~ since all angles are 60. c: Yes, zoom factor of 2.5; translate so that one pair of corresponding vertices coincide, rotate so that rays coincide, and dilate. d: No, since corresponding angles are not equal. Note that you cannot apply zoom factor to angles a: One strategy: Translate one so that the centers coincide. Then dilate so that the radius is the same as the other circle. b: Equilateral triangles, which from part (b) of 3-54 were similar because they have equal angle measures. Squares or other regular polygons are also always similar a: There are 12 combinations. One way to systematically list them all is to list a bus number (such as 41) and then match it with each possible activity. This can be repeated for each of the possible bus numbers. b: i: 9 12, ii: 8 12, iii: See graph at right. Perimeter = 44.9 units; Area = 94 sq. units y a: ABCD ~ EVOL b: RIGHT ~ RONGW c: One possible answer: ΔTAC ~ ΔGDO x 16 Core Connections Geometry

17 Lesson a: x = 20 mm b: w = 91 mm a: Impossible: can be rejected using Triangle Inequality or Pythagorean Theorem. b: Possible c: Impossible: rejected because the sum of the angles is a: 8 12 b: This reasoning is incorrect. Rewrite it is raining in the lower left oval, and Andrea s flowers must be closed up in the right oval a: Reflection, rotation, and translation b: Rotation and translation c: Rotation, dilated by factor of 2, and translation a: Possible b: Not possible because the sum of the measures of an obtuse and right angle is more than 180. c: Not possible because a triangle with sides of equal length obviously cannot have sides of different lengths. d: Possible Selected Answers 17

18 Lesson a: (5, 2) b: ( 4, 2) c: (3, 3); It is the center of the figure, or the midpoint of each diagonal a: y = 1 2 x + 2 b: A = 4 sq. units; P = ! units c: y =!2x a: x = 51 alternate interior angles and Triangle Sum Theorem b: x = 43 circle has 360º c: x = 1 Pythagorean Theorem a: See tree diagram at right. b: Yes c: 1 6, 6 3 d: 1 2 e: 2 6, no the spinners are independent, because now the possible outcomes are $100, $200, $1500, $200, $400, and $ a: n = 32 b: m! $100 $300 $1500 double $200 keep $100 double $600 keep $300 double $3000 keep $ Missing side length of first rectangle must be 4 m because the perimeter is 26 m. Missing side length of second rectangle must be 9" because the area is 36 sq. in. Since angles are equal and ratios of corresponding side lengths are equal, therefore, the rectangles are similar. In fact, they are congruent because r = Core Connections Geometry

19 Lesson a: Scalene triangle b: Isosceles triangle c: Not possible d: Equilateral triangle a: The two equations should have the same slope but a different y-intercept. This forces the lines to be parallel and not intersect. b: When solving a system of equations that has no solution, the equations combine to create an impossible equality, such as 3 = 0. Another special case occurs when the resulting equality is always true, such as 2 = 2. This is the result when the two lines coincide, creating infinite points of intersection a: Not similar, interior angles are different. b: Must be similar by AA ~. c: Similar, all side lengths have the same ratio Perimeter = = 38 units, height of triangle = 8 units, area = 60 square units This reasoning is correct a: 3(4x!12) = 180, x = 18 b: 4.9 2! = x 2, x! 3.79 c: x + (180! 51!103 ) + 82 = 180, x = 72! d: 3x! 2 = 2x + 9, x = 11 Selected Answers 19

20 Lesson a: SSS ~ and SAS ~ (if you show that the triangles are right triangles) b: AA ~ and SAS ~ c: None since there is not enough information a: = 60% b: 18 x = 3 10, x = a: 12x 2! 7x!10 b: 16x 2! 8x +1 b: x =! 5 9 c: x = !y = 48! because of vertical angles;!z = 48! because of reflection of!y or because of angle of incidence = angle of reflection with!x a:! 5 6 b: LD = 61! 7.81units c: Calculate x and y by determining the difference in the corresponding coordinates Original: A = 135 sq. units, P = 48 units; New: A = 15 sq. units, P = 16 units Lesson x = 137,!y = h = 5 feet; perimeter 24.2 feet a: 28? = 2 5 ; There are 70 animals in the bin. b: c: 3? = = 40% = 5% ; You need a total of 60 animals in the bin a: y =! 1 2 x + 4 b: y = 2x!1 c: y = 2 5 x d: C = (t!1) = 8 + 7t miles Possible response: Rotate WXYZ clockwise, translate it to the left, and dilate it by a factor of 0.4. y = 7.5,!z = Core Connections Geometry

21 Lesson a: x = 11 b: x = 45 c: x = 30 d: x = a: See flowchart at right. b: Yes, because the triangles are similar (AA ~) and the ratio of the corresponding side lengths is 1 (because AC = DF) a: Yes, she used the Pythagorean Theorem. m A = m D b: (x +1) 2 = x 2 + 2x +1 c: x = 24 d: 56 units ΔABC ~ ΔDEF m B = m E AA ~ 4-9. x = 9,!y = 4,!z = a: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. b: Yes. c: P(even) = ; P(10) = 3 36 ; P(15) = 0 d: The sum of 7. P(7) = 6 36 = If h represents the number of hours and t represents the temperature, then t = h and t = 92! 2h ; h = 3 hours and the temperature will be 86 F. Lesson a:! = 11, x 95! 1 5, x! b: a = b = 45 c: y 70! 5 2, y! a: side ratio = 4:1 b: perimeter ratio is 4:1 c: 28' a: Yes, AA ~. b: No, side ratios not equal 12 64! c: Cannot tell, not enough angle values given ! 3 2 = 27, 9 2! 3 2 = 72. So perimeter is ! area is ( )(3) 2! sq. cm. cm. The Since the slope ratio for , AB 50 feet. The slope ratio for , so BC 4 feet. Thus, AB is actually longer a: 12 b: Yes c: 6 12 = 1 2 ; 8 12 = 2 3 Selected Answers 21

22 Lesson They both could be. It depends on which angle is used as the slope angle a: Yes, since the slope ratio is greater than 1, the angle must be greater than 45. b: Isiah is correct. Since the angle is less than 45, the slope ratio must be less than 1. c: Since the angle is greater than 45, x must be less than a: A; a n = 1+ 3(n!1) = 3n! 2 b: neither c: G; a n = 2!2 n"1 = 2 n d: A; a n = 5 + 7(n!1) = 7n! Answers vary, possible solution: square, equilateral triangle, and equilateral hexagon a: 2 5 b: Yes. If the first song is a country song, then there is only 1 country song left to play out of 4 songs. Therefore, the chance that the second song is a country song is 1 4. c: 1 2, because only 2 songs are left and only one is sung by Sapphire. d: 1 3 ; same m ABC = 22, m BAC = 68, sum = 90 ; complementary 22 Core Connections Geometry

23 Lesson a: t = b: p! c: b! a: x = 30 2 ; x = 24 b: 2x x x + 2x = 360, x = 40 c: 5 12 = 3 x, x = 36 5 = They are congruent. Possible response: Reflect ΔADS across a vertical line, then translate it possible ways: ABCD, ABDC, ACBD, ACDB, ADBC, ADCB, BACD, BADC, BCAD, BCDA, BDAC, BDCA, CABD, CADB, CBAD, CBDA, CDAB, CDBA, DABC, DACB, DBAC, DBCA, DCAB, DCBA Her father s eyes were inches high a: A = 144 cm 2, P = 52 cm b: A = m 2, P = m c: A = 72 sq cm, P = 48 cm d: A = 130 sq. feet, P = 58 feet Lesson a: Either 3 or 1 3 b: Either 9 or a: 3x x + 7 = 90, x = 20 b: 9x + 4 = 3x +14 ; x = 10 6! (a) and (d) are most likely independent b: feet a: G; a n = 1 2! 1 n"1 2 = 1 n 2 b: A; a n =!7.5! 2(n!1) = 5.5! 2n (x + 3) 2 = 26 2 ; x = 21 Selected Answers 23

24 Lesson a: = 1 2 b: 9 19 c: No, they are not independent. The probability the second contestant is a girl depends on whether the first contestant was a girl or not See graph at right. a: (! 1 2,!0) and (3,!0) b: x =! 1 2 or x = 3 ; Yes a: Slope = 1 2 b: It must be parallel to or coincide with the line graphed at right Francis: y = x + 2, John: y = 3 4 x + 5 ; 12 seconds a: x b: x c: x = 1.5 or 5 d: No solution leg units, hypotenuse cm, so the perimeter cm Lesson a: A tree diagram; a third dimension would be needed to represent the three coins with an area model. b: See tree diagram at right; 8 c: i: 1 8, ii: 3 8, iii: 7 8, iv: 3 8 d: They are both the same probability of 50%. e: The sample space remains the same; i: , ii: = , iii: , iv: H T H H T H T T H H T T H T Yes, they are similar due to AA ~ because m B = m E and m C = m C (triangles share an angle) ; If the die is fair, each roll of the die is an independent event a: It implies that because Brian is always late on Tuesday, then today must be Tuesday. b: The Brian is always late on Tuesdays and Today is Tuesday ovals should be next to each other, both with arrows pointing to Brian will be late today a: 3, 15, 75, 375 b: 10, 50, 250, x! 10.39, y = Core Connections Geometry

25 Lesson Both equal y = 1 3 x a: See diagram at right. b: Ratio for tan , so 170 x! 1 5, and x! 850 feet. Alternatively, a calculator could be used and x = tan(11 ) 170! 875 feet. 11º x 170 ft a: x = 49 b: x = 2 c: x = 16 3 d: x =!5 or No. Triangle Inequality property prevents this because < 20 and 20!10 > $450 Lesson a: 3 36 b: 4 36 c: a: 5 ways b: 6 ways c: 11 d: a: x = 13, Pythagorean Theorem b: x = 80, Alternate interior angles and the Triangle Angle Sum (x + 2)(x + 5) = 40, x 2 + 7x! 30 = 0, so x =!10 or 3. Since x cannot be negative, x = 3. Therefore, the dimensions of the rectangle are 5 and 8 units a: Less than 45 b: Equal to 45 c: More than The slope is! Points will vary. y =!0.7x A few possible solutions: (5, 79), (15, 72), (25, 65), etc. Selected Answers 25

26 Lesson a: 1 12 b: Intersection c: No, P(yellow) = 1 6 d: 2 3 e: You cannot move = 2 3 or you can move 1 3 of the time and 1! 1 3 = a: y = 3 b: y = It assumes that everyone who likes bananas is a monkey feet " < x < 14 " Methods vary: θ = 68º (could be found using corresponding and supplementary angles), α = 85º (could be found using corresponding angles since lines are parallel a: P(K) = 52 4, P(Q) = 52 4, P(C) = b: ; You can add the probabilities of king and club, but you need to subtract the number of cards that are both kings and clubs (1). P(K or C) = ! 52 1 = c: P(K or Q) = 52 8 = 2. There is no overlap in the events so you can just add the probabilities. 13 d: P(not a face card) = 1! = years a: Yes, ΔABD ~ ΔEBC by AA ~. b: Yes. Since DB = 9 units (by the Pythagorean Thm), the common ratio is LE = MS and LI = ES = MI AB! mm, A! sq. mm a: A ( 3, 3), B (9, 3), C ( 3, 6) b: A ( 3, 3), B ( 3, 9), C ( 6, 3) c: (9, 3) 26 Core Connections Geometry

27 Lesson x 7.50 and y 8.04 units; Use either sine or cosine to get the first leg, then any one of the trig ratios or the Pythagorean Theorem to get the other a: False (a rhombus and square are counterexamples) 5-9. B a: b: True c: False (it does not mention that the lines must be parallel) (4 cards less than 5)! (4 suits) 52 = If Aces are not included, 52. b: 1! = If Aces are included, 52. c: P(red) + P(face) P(red and face) = ! 52 6 = area = 74 sq ft, perimeter = ft a: x = 3 b: m = 10 c: p =!4 or 2 3 d: x = 23 Lesson a: sin 22 = x 17 b: x! 6.37, tan 49 = 7 x, 6.09 c: cos 60 = x 6, x = feet a: G; a n = 100( 1 10 )n!1 = 10 3!n b: A; a n = 0! 50(n!1) = 50! 50n Region A is 1 4 of the circle. Since the spinners are independent, the probability of A and A is 1 4! 1 4 = In 80 games, we expect A and A to occur 16 (80) = 5 times a: False (a triangle is a counterexample) b: False (this is only true for rectangles and parallelograms) c: True a: 6x 2! x! 2 b: 6x 3! x 2!12x! 5 c:!3xy + 3y 2 + 8x! 8y d: x 2! 9y ΔABC ~ ΔEFD by SAS ~ Selected Answers 27

28 Lesson a: x = ±5 b: All numbers c: x = 2 d: No solution Using cos A = 13 5, sin A = 12 13, or tan A = 12 5, A! 67.4º 5-31.!11.5 seconds a: 1 36 b: Area sq m, perimeter m a: It uses circular logic. b: Reverse the arrow between Marcy likes chocolate and Marcy likes Whizzbangs. Also, remove the arrow connecting Marcy likes chocolate and Whizzbangs are 100% chocolate minutes Lesson All of the triangles are similar. They are all equilateral triangles Since tan(33.7 )! 2 3, y! 2 3 x a: a 1 = 108, a n+1 = a n +12 b: a 1 = 2 5, a n+1 = 2!a n c: a n = 3741! 39(n!1) = 3780! 39n d: a n = 117(0.2) n!1 = 585(0.2) n a: sin! = b a b: tan! = a b c: cos! = a b a: ; union b: 3 52 ; intersection c: 1! = a: cos!23 = 18 x or = 18 x b: Since 67 is complementary to 23, then sin 67 = cos 23. So sin 67! Core Connections Geometry

29 Lesson a: A = 1 sq. m, P = m b: A = ! square ft, P = ! ft a: y = 111º ;!x = 53º b: y = 79º ;!x = 47º c: y = 83º ;!x = 53º d: y = 3;!x = 3 2 units a: 4 2 units; Use the Pythagorean Theorem or that it is a triangle. b: It is a trapezoid; 24 square units % by using the Addition Rule a: Answers vary; sample responses: x < 3, x is even, etc. b: The length of each leg is 6 units a: a n = (n!1) = 1500n!1000 b: a n = 30(5) n!1 = 6(5) n a: Not similar b: SAS ~ c: SSS ~ d: SSS ~ or SAS ~ Selected Answers 29

30 Lesson a: 16 inches b: 4 yards and 4 2 yards c: 24 feet d: 10 meters and 10 3 meters a: m A = 35º, m B = 35º, m ACB = 110º, m D = 35º, m E = 35º, m DCE = 110º b: Answers vary. Once all the angles have been found, state that two pairs of corresponding angles have equal measure, such as m A = m D and m B = m E to reach the conclusion that ΔABC ~ ΔDEC by AA ~ or AC = BC, DC = EC, and m ACB = m DCE by SAS ~. c: They are both correct. Since both triangles are isosceles, we cannot tell if one is the reflection or the rotation of the other (after dilation) cos!52 = b c, tan!52 = a b, cos!38 = a c, sin! = 40 x, x! inches a: explicit b: a n =!3+ 4(n!1) = 4n! 7 c: a 50 = 193 d: a n = 3! 1 3 (n!1) = 3 1 3! 1 3 n $3( ) + $5( ) + (!$6)( ) = $1.50. It is not fair because the expected value is not a: 1 2 b: 0 c: 3 4 d: 1 Lesson a: Impossible because a leg is longer than the hypotenuse. b: Impossible because the sum of the angles is more than William is correct a: A! ( 3, 6), B! ( 5, 4), C! (0, 4) b: A!! (3, 3), B!! (1, 1), C!! (1, 6) a: x = 16 5 b: No solution c: x = 11 or 3 d: x = b is correct; if two sides of a triangle are congruent, the angles opposite them must be equal. 30 Core Connections Geometry

31 Lesson They must have equal length. Since a side opposite a larger angle must be longer than a side opposite a smaller angle, sides opposite equal angles must be the same length mm square units See tree diagram at right (an area model is not practical). P(three yogurts) = 12.5%. 100%!12.5% = 87.5% chance of not getting three yogurts a: x = 45 4 = b: x =!10 or x = 10 c: x = (!2,!4) d: No solution yogurt green apple red apple yogurt green apple red apple yogurt green apple red apple yogurt green apple red apple yogurt green apple red apple yogurt green apple red apple yogurt green apple red apple yogurt green apple red apple yogurt green apple red apple yogurt green apple red apple yogurt green apple red apple yogurt green apple red apple yogurt green apple red apple Selected Answers 31

32 Lesson a: 29 b: cos!29 = y 42, y! a: ( 1, 2) b: (4, 4) c: (3, 4) tan!1 ( 3 4 ) " sin!1 7 8 " 61.0º a: 2x 2 + 6x b: 3x 2! 7x! 6 c: x = 1 or 7 d: y = 3 or a: 1 12 b: 1 3 Lesson a: The diagram should be a triangle with sides marked 116 ft. and 224 ft. and the angle between them marked 58. b: 190 feet, Law of Cosines a: Corresponding angles have equal measure. b: The ratio of corresponding sides is constant, so corresponding sides are proportional y = (tan 25 )x + 4 or y! 0.466x It must be longer than 5 and shorter than 23 units terms (3) (!1) (10) = 11 6 " $1.83 The game is not fair because the expected value is not zero years 32 Core Connections Geometry

33 Lesson The third side is 12.2 units long. The angle opposite the side of length 10 is approximately 35.45, while the angle opposite the side of length 17 is approximately x! 11.3 units; Methods include using the Pythagorean Theorem to set up the equation x 2 + x 2 = 16 2, using the triangle shortcut to divide 16 by 2, or to use sine or cosine to solve using a trigonometric ratio No, because to be a rectangle, the parallelogram needs to have 4 right angles. Counterexample: A parallelogram without 4 right angles a: P mm, A = 72 sq mm b: P = 30 feet, A = 36 square feet A(2, 4), B(6, 2), C(4, 5) The expected value per throw is 1 4 (2) (3) (5) = 15 4 = 3.75, so her expected winnings over 3 games are 3(3.75) = 11.25; yes, she should win enough tickets to get the panda bear y = 3 4 x a: m ABE = 80º, m EBC = 60º, m BCE = 40º, m ECD = 80º, m DEC = 40º, m CEB = 80º, m BEA = 60º b: a: 8.64 cm b: PS = SR = 5.27 cm, so the perimeter is 25.5 cm Area sq. units, perimeter units a: Explicit t(n) =!2 + 3n ; Recursive t(0) =!2,!t(n +1) = t(n) + 3 b: Explicit t(n) = 6( 1 2 )n ; Recursive t(0) = 6,!t(n +1) = 1 2 t(n) c: t(n) = 24! 7n d: t(n) = 5(1.2) n e: t(4) = 1620 Chain a: See diagram at right. b: x = ! 5.77 Bush x x Shed a: ! units b:! c: ( 2, 0) Selected Answers 33

34 Lesson a: Alternate interior angles. b: Vertical angles. c: u & z, s & x, v & w, and t & y 6-5. a: They are similar by SAS ~. b: Yes, because they are similar and the corresponding sides have a ratio of x = 180, x = 127 3! a: 8 cm b: ft c: 15.2 in a and 1b: statements ii and iv, 2: The cupcakes are burned, 3: The fans will not buy the cupcakes because they are burned, 4: The team will not have enough money for uniforms a: A or 75% b: 3 20 or 15% c: 1 or 100% c: (b) is an intersection, and (c) is a union. Lesson a = 97º, b = 15º, c = 68º, d = 68º a: 3.75, tangent b: 7 2! 9.9, Pythagorean Theorem or ratios c: 9.54, Law of Cosines a: 25 units b: 56 sq. units and 350 sq. units a: A ( 2, 7), B ( 5, 8), C ( 3, 1) b: A (2, 7), B (5, 8), C (3, 1) c: Reflection across the y-axis a: (0.8)(0.8) = 0.64 = 64% b: (0.8)(0.2) = 0.16 = 16% A 34 Core Connections Geometry

35 Lesson a: Not similar because there are not three pairs of corresponding angles that are congruent. b: Similar (AA ~) a: y = 5 2 x! 8 b: y = 3 2 x b: = DE 10, DE a: Yes because of AAS or ASA ; ΔDEF ΔLJK. b: One possible answer, a reflection across line segment JK and then a translation of ΔDEF to line up point J and point E, followed by a rotation. c: KL 4.3 units c = 10 by substitution a: P(A or B) = P(A) + P(B) P(A and B) = 64 of A and B (the overlap) was ! 0 = " 83.0% ; the probability b: P(A or B) = P(A) + P(B) P(A and B)! 75% = ! x " x # 5.1% Selected Answers 35

36 Lesson Reasoning can vary. See sample responses below. a: a = 123, when lines are //, corr. s are =, b = 123, when lines are //, alt. int. s are =, c = 57, suppl. s b: all = 98, suppl. s, then when lines are //, alt. int. s = and corres. or vert. s = c: g = h = 75, when lines are //, alt. int. or corres. s =, then vert. s = a: Similar (SSS ~) b: Similar (AA ~) a: x =!4 and y = 0 b: No solution; the lines are parallel = 5 x+5, x = Let B represent the measure of angle B. Then (3B + 5º) + B + (B 20º) = 180º, so m A = 122º, m B = 39º, and m C = 19º a: See possible area model at right C b: 1 4 c: = 25 36! 69% parents 1 3 niece 1 6 boyfriend 1 2 parents 1 3 niece boyfriend Core Connections Geometry

37 Lesson Justifications and order may vary: a = 53, given; b = 55, straight angle (with g); c = 72, triangle angle sum; d = 53, when lines are parallel, alternate interior angles are equal; e = 55, when lines are parallel, alternate interior angles are equal; f = 127, straight angle (with a), so they are supplementary a: For left-hand triangle: c 2 = ! cos 60, c = 3 3! units; For right-hand triangle: c 2 = ! cos 30, c = 3 units; They are congruent. b: Yes; by SSS or SAS a: Converse: If the ground is wet, then it is raining. Not always true. b: Converse: If a polygon is a rectangle, then it is a square. Not always true. c: Converse: If a polygon has four 90 angles, then it is a rectangle. Not always true. d: Converse: If a polygon is a triangle, then it has three angles. Always true. e: Converse: If vertical angles are congruent, then two lines intersect. Always true x-intercept: (4, 0), y-intercept: (0, 6) a: y = 13 4 b: y = 2 c: " d: x = a: 8 3 b: 1 8 c: 8 3 d: 1 8 ; sum must be equal to one sin 40 = 600 h, h feet Selected Answers 37

38 Lesson a: x! b: x! c: x! d: x = square units; First find AC = 5 and then calculate 1 2 (5)(3.6), or use BC as the base and calculate 1 2 (2 + 4)(3) a: m = 33 m, n = 36 m b: Area (small) = 378 cm 2, perimeter (small) = 80 cm, area (big) = m 2, and perimeter (big) = 120 m a: Similar because of AA ~. b: Neither because angles are not equal. c: Congruent because of ASA or AAS a: b: y = x + 3 c: (1, 4) D 38 Core Connections Geometry

39 Lesson a: Lines l and m are parallel because alternate interior angles are equal. b: Line n is perpendicular to line m because w + k = 180 and if w = k, then each is 90. c: No special statements can be made because vertical lines are always equal. d: Lines l and m cannot be parallel because otherwise z + k = a: ΔABC ~ ΔDEF (AA ~) b: ΔMON ΔPQR, (AAS or ASA ) c: Neither congruent nor similar because m!j " 62!. If m!j = 62!, then m!l = 180! " 2 #62! = 56!. Since sin 56 5! sin 72 8, this triangle cannot exist a: Converse: If the cat runs away frightened, then it knocked over the lamp. Not always true. b: Converse: If the chances of getting a 3 are 1 6 always true., then a 6-sided dice was rolled. Not c: Converse: If a triangle is a right triangle, then it has a 90 angle. Always true D!##"!##" a: It is a trapezoid. The slope of WZ equals the slope of XY. b: 18.3 units c: ( 9, 1) d: 2 Selected Answers 39

40 Lesson a: Yes, because parallel lines assure us that the alternate interior angles are congruent. Since corresponding angles in the triangles have equal measure, the triangles are similar by AA ~. b: x 20 = x+2 24, x = a: x = 4 b: x = 55 c: x = 23 and y = 43 d: x = 5.5 and y = P(A or B) = P(A) + P(B) P(A and B) = 4% %! 1 2 % = 4%. If a refrigerator has a dent it also always has a paint blemish area sq. yards; perimeter yards a: 288 feet by 256 feet C b: area of shape = 59.5 square units; area of island = 60,928 square feet Lesson a: Congruent (HL or SAS ) b: Congruent (AAS ) c: Not necessarily congruent. d: Congruent (SAS ) a: x + 4x! 2 = 90, x = 18.4, complementary angles b: 2m m!1 + m + 9 = 180, m = 42.25, Triangle Angle Sum Theorem c: 7k! 6 = 3k +18, k = 6, vertical angles are equal d: 16 x = 13 8, x! 9.8, corresponding parts of similar figures have equivalent ratios x = 11; m ABC = 114º a: Converse: If a triangle is isosceles, then its base angles are congruent. Always true % D b: Converse: If the sum of the angles in a figure is 180, then the figure is a triangle. Always true. c: Converse: If my mom is happy, then I cleaned my room. Not always true. 40 Core Connections Geometry

41 Lesson a: 5x + 3 = 4x + 9 because if lines are parallel, then alternate interior angles are equal, x = 6º. b: q = t because if lines are parallel, then corresponding angles are equal; c + t = 180º because if lines are parallel, then same side interior angles are supplementary; 66 c: = 92 ; g + q = 180 because when lines are parallel, same-side interior angles are supplementary a: y = 6 5 x! 3 b: y =! 1 4 x c: y = 1 3 x d: y = a: x! 8.1 b: Not enough information. c: x! a: x = 15, Triangle Angle Sum b: k = 5, Isosceles triangle c: t = 9 and w = 131, parallel lines d: x! 49.94, Triangle Angle Sum a: $1.50 b: $ B Lesson a: They are congruent by ASA or AAS. b: AC 9.4 units and DF = units 7-7. Relationships used will vary, but may include alternate interior angles, Triangle Angle Sum Theorem, etc.; a = 26, b = 65, c = 26, d = width = 60 mm, area = 660 mm A quadrilateral a: x = 18, y = 9 3 b: x = 24 2, y = 24 c: x = 8 = , y = 16 = a: (6, 13) b: Not possible, these curves do not intersect. Selected Answers 41

42 Lesson Using the Pythagorean Theorem, AB = 8 and JH = 5. Then, since 3 6 = 4 8 = 5 10, ΔABC ~ ΔHGJ because of SSS ~ cm Line L: y =! 1 6 x + 6 ; line M: y = 2 3 x +1; point of intersection: (6, 5) a: 3m = 5m! 28, m = 14 b: 3x x! 8 = 180, x = 15 c: 2(n + 4) = 3n!1, n = 9 units d: 2(3x +12) = 11x!1, x = 5 units Rotating about the midpoint of a base forms a hexagon (one convex and one nonconvex). Rotating the trapezoid about the midpoint of either of the non-base sides forms a parallelogram a: 10 units b: ( 1, 4) c: 5 units, it must be half of AB because C is the midpoint of AB. Lesson (Day 1) a: The 90 angle is reflected, so m XYZ = 90º. Then m YZY = 180º M(0, 7) b: They must be congruent because rigid transformations (such as reflection) do not alter shape or size of an object. c: XY! X Y ", XZ! XZ, YZ! Y " Z, Y Y, YXZ Y XZ, and YZX Y ZX c 2 and a 2 + b a: The triangles are similar by SSS ~. b: The triangles are similar by AA ~. c: Not enough information is provided. d: The triangles are congruent by AAS or ASA a: It is a parallelogram; opposite sides are parallel. b: 63.4 ; They are equal. c: AC :!y = 1 2 x + 1 2, BD :!y =!x + 5 ; No d: (3, 2) a: No solution, lines are parallel. b: (0, 3) and (4, 11) 42 Core Connections Geometry

43 Lesson (Day 2) Side length = 50 units; diagonal is 50!! 2 = 100 = 10 units a: It is a rhombus. It has four sides of length 5 units. b: HJ : y =!2x + 8 and GI : y = 1 2 x + 3 c: They are perpendicular. d: (6, 1) e: 20 square units a: P(scalene) = 1 4 b: P(isosceles) = 2 4 = 1 2 c: P(side of the triangle is 6 cm) = 2 4 = a: 6n! 3 = n +17, n = 4 b: 7x!19 + 3x +14 = 180, so x = Then 5y! 2 = 7(18.5)!19, so y = 22.5 c: 5w w = 180, w = 18 d: k 2 = ! 2(15)(25) cos120, k = The graph is a parabola with roots ( 3, 0) and (1, 0), and y-intercept at (0, 3) m Selected Answers 43

44 Lesson a: = 10 sides b: Regular decagon If the diagonals intersect at E, then BE = 12 mm, since the diagonals are perpendicular bisectors. Then ΔABE is a right triangle and AE = 15 2!12 2 = 9 mm. Thus, AC = 18 mm Yes, she is correct. Show that the lengths on both sides of the midpoint are equal and that (2, 4) lies on the line that connects ( 3, 5) and (7, 3) a: See flowchart at right. b: Not similar because corresponding sides do not have the same ratio. c: See below. AA ~ ΔFED ~ ΔBUG SSS ~ (a) and (c) are correct because if the triangles are congruent, then corresponding parts are congruent. Since alternate interior angles are congruent, then AB // DE AB = 40! 6.32, BC = 34! 5.83, therefore C is closer to B. 44 Core Connections Geometry

45 Lesson a: x = 8.5 b: x = 11 c: x = a: = 5 sides b: = feet from the point on the street closest to the art museum a: a n = n = (n!1) b: a n = 6( 1 2 )n = 3( 1 2 )n! a: (0.7)(0.7) = 0.49 = 49% b: (0.3)(0.7) = 0.31 = 21% a: Similar (SSS ~) b: Congruent (ASA or AAS ) c: Congruent, because if the Pythagorean Theorem is used to solve for each unknown side, then 3 pairs of corresponding sides are congruent; thus, the triangles are congruent by SSS ). d: Similar (AA ~) but not congruent since the two sides of length 12 are not corresponding Possible response: Rotate the second triangle 180 and then translate it to match the sides with the first triangle. Selected Answers 45

46 Lesson x!1 = x + 8, x = 3 ; 5y + 2 = 22, y = a: 0.8 b: 1200(0.8) 3 = $ c: 1200(0.8)!2 = $ a:!a = 36,!r = 54,!m = 54,!y = 72,!z = 108 b: Possible response:!y and!z are supplementary interior same side angles a: It is a parallelogram, because MN // PQ and NP // MQ. b: (1, 5) E is a midpoint Given Vertical angles are congruent. AEB CED Definition of midpoint. Definition of midpoint. ΔAEB ΔCED SAS a: 50%; The sum must be 100% b: central angle for red = 0.4(360 ) = 144, white = 0.1(360 ) = 36, blue = 0.5(360 ) = 180 c: Yes; there could be more than three sections to the spinner, but the ratio of the areas for each color must match the ratios for the spinner in part (b). 46 Core Connections Geometry

47 Lesson a: Congruent (SSS ) b: Not enough information c: Congruent (ASA ) d: Congruent (HL ) See answers in problem a: 83 b: a: Yes; HL b: 18, 4 c: tan18 = AD 4, AD 12.3 units d: 49.2 square units a: Parallelogram because the opposite sides are parallel. b:! AC ##" : y = 4 3 x ;! BD ##" : y =! 2 3 x a: , since 64 = 8, then 68 must be a little greater. b: (1) 2.2, (2) 9.2, (3) 7.1, (4) a: 2x + 52 = 180, 64 b: 4x! 3 + 3x +1 = 180, 26 c: sin 77 x = sin 72 8, x 8.2 d: 5x + 6 = 2x + 21, x = 5 Lesson ! 62.4 square units No. Using the Pythagorean Theorem and the Law of Cosines, the perimeter of the triangle is 26.3 feet a: x cm, tangent b: x 7.86 mi, Law of Sines c: x 15.3', Law of Cosines a: Congruent (SAS ) and x = 2 b: Congruent (HL ) and x = A = 24 square units a: 20 4 = 1 5 b: 4 5 ; Since the sum of the probabilities of finding the ring and not finding the ring is 1, you can subtract 1! 1 5 = 4 5. c: No, his probability is still 20 4 = 1 5 because the ratio of the shaded region to the whole sandbox is unchanged a 50 =!130 Selected Answers 47

48 Lesson a: See diagram at right. b: Since corresponding parts of congruent triangles are congruent, 2y + 7 = 21 and y = y-intercept: (0, 6), x-intercept: (4, 0) A T 21 P m a = 132º, m b = 108º, and m c = 120º, m a + m b + m c = 360º % AB! CD and AB // CD (given), so BAC DCA (alt. int. angles). AC! CA (Reflexive Property) so ΔABC ΔCDA (SAS ). BCA DAC (!!!s!"!!!parts ). Thus, BC // AD (if alt. int. angles are congruent, then the lines cut by the transversal are congruent) Because alternate interior angles are congruent, the angle of depression equals the angle formed by the line of sight and the ground. Then tan!! = 52 38,! " a: ΔADC; AAS or ASA b: ΔSQR; HL c: No solution, only angles are congruent. d: ΔTZY; SAS and vertical angles e: ΔGFE; alternate interior angles equal and ASA f: ΔDEF; SSS 48 Core Connections Geometry

49 Lesson ZY! WX, YZX WXZ, ZYW YWX (alt. int.! s), ΔXWM ΔZYM, ASA!, YM! WM and XM! ZM,! " #! parts Typical responses: right angles, congruent diagonals, 2 pairs of opposite sides that are congruent, all sides congruent, congruent adjacent sides, diagonals that bisect each other, congruent angles, etc a: The triangles should be by SSS but 80º 50º. b: The triangles should be by SAS but 80º 90º and 40º 50º. c: The triangles should be by SAS but d: Triangle is isosceles but the base angles are not equal. e: The large triangle is isosceles but base angles are not equal. f: The triangles should be by SAS but sides The triangles described in (a), (b), and (d) are isosceles a: 12 b: 15 c: See reasons in bold below. Statements 1. AD // EH and BF // CG Given Reasons 2. a = b If lines are parallel, alternate interior angle measures are equal. 3. b = c If lines are parallel, corresponding angle measures are equal. 4. a = c Substitution 5. c = d Vertical angle measures are equal. 6. a = d Substitution This problem is similar to the Interior Design problem (7-21). Her sink should be located feet from the right front edge of the counter. This will make the perimeter 25.6 feet, which will meet industry standards. Selected Answers 49

50 Lesson a: (4.5, 3) b: ( 3, 1.5) c: (1.5, 2) a: ΔSHR ~ ΔSAK because ΔSHR can be dilated by a factor of 2. b: 2HR = AK, 2SH = SA, SH = HA c: 6 units a: 4 b: 1 c: a: ΔCED; vertical angles are equal, ASA b: ΔEFG; SAS c: ΔHJK; HI + IJ = LK + KJ, J J, SAS. d: Not, all corresponding pairs of angles equal is not sufficient See answers in problem No. Her conclusion in Statement #3 depends on Statement #4, and thus must follow it a: Must be a quadrilateral with all four sides of equal length. b: Must be a quadrilateral with one pair of opposite sides that are parallel. 50 Core Connections Geometry

51 Lesson a: Yes, each side has the same length ( 29 units). See graph at right. b:! BD ##" is y = x ;! AC ##" is y = 5! x c: The slopes are 1 and 1. Therefore the diagonals are perpendicular Multiple answers are possible. Any order is valid as long as Statement #1 is first, Statement #6 is last, and Statement #4 follows both Statements #2 and #3. Statements #2, #3, and #5 are independent of each other and can be in any order as long as #2 and #3 follow Statement # a: 6 b: 3 c: a: Yes, by SAS ~. b: FGH FIJ, FHG FJI c: Yes, because corresponding angles are congruent. (Triangle Midsegment Theorem) d: 2(4x! 3) = 3x +14, so x = 4 and GH = 4(4)! 3 = 13 units a: It is a right triangle because the slopes of AB and AC ( 4 3 and! 4 3, respectively) are opposite reciprocals. b: B! is at ( 2, 0). It is an isosceles triangle because B AB must be a straight angle (because it is composed of two adjacent right angles) and because BC! B" C (because reflections preserve length) a: feet b: x 7 yds c: x d: feet a: Must be: trapezoid. Could be: isosceles trapezoid, parallelogram, rhombus, rectangle, and square. b: Must be: parallelogram. Could be: rhombus, rectangle, and square. Selected Answers 51

52 Lesson a: (8, 8) b: (6.5, 6) a: X and Y b: Y and Z a: Must be: none. Could be: right trapezoid, rectangle, square. b: Must be: none. Could be: kite, rhombus, square a: If a polygon is a parallelogram, then its area equals its base times its height. b: If the area of a polygon is the base times its height, then the polygon is a parallelogram. This is always true. c: If a polygon is a triangle, then its area equals one half its base times its height. Arrow diagram: Polygon is a triangle area of the polygon equals one-half base times height. d: If a polygon s area is one-half its base times its height, then the polygon is a triangle. This is always true a: = 20 sides b: It can measure 90 (which forms a square). It cannot be 180 (because this polygon would only have 2 sides) or 13 (because 13 does not divide evenly into 360 ) The expected value is 1 4 ($3) ($1) = $1.50 per spin, so each player should pay $1.50 so that there is no net gain or loss over many games a: They are perpendicular because their slopes are opposite reciprocals. b: It could be a kite, rhombus, or square because the diagonals are perpendicular and at least one diagonal is bisected. 52 Core Connections Geometry

53 Lesson a: 110 b: 70 c: 48 d: a: no b: yes c: no d: yes e: no f: yes g: yes h: no 8-8. b: The measure of an exterior angle of a triangle equals the sum of the measures of its remote interior angles. c: a + b + c = 180º (the sum of the interior angles of a triangle is 180 ), x + c = 180º (straight angle); therefore, a + b + c = x + c (substitution) and a + b = x (subtracting c from both sides) x = 72 and y = = a: congruent (SAS ), x = 79 b: Cannot be determined. c: congruent (AAS ), x 5.9 units d: congruent (SAS ), x a: True b: False (counterexample is a quadrilateral without parallel sides.) c: True d: True e: False (counterexample is a parallelogram that is not a rhombus) Selected Answers 53

54 Lesson a: Isosceles right triangle, because AC = BC and AC! BC. b: 45 ; methods vary a = 87, b = 83, c = 96, d = 94 ; A = 40.5 square miles, P 27.7 miles ( 5, 1), ( 3, 7), and ( 6, 2) a: a n =!45 +15n =!30 +15(n!1) b: a n = 27( 1 3 )n = 9( 1 3 )n! See answers in bold below B Statements 1. BC // EF, AB // DE, and AF = DC 1. Given 2. m BCF = m EFC and m EDF = m CAB Reasons 2. If two lines cut by a transversal are parallel, then alternate interior angles are equal. 3. FC = FC 3. Reflexive Property 4. AF + FC + CD = FC 4. Additive Property of Equality (adding the same amount to both sides of an equation keeps the equation true) 5. AC = DF 5. Segment addition 6. ΔABC ΔDEF 6. ASA 7. BC! EF 7.!!!s!"!!!parts 54 Core Connections Geometry

55 Lesson a: A = 36 sq. ft, P = 28 ft b: A = 600 sq. cm, P cm QP = RS and PR = SQ (given), QR = QR (Reflexive Property), so ΔPQR ΔSRQ (SSS ) and P S ( Δs parts) a: Isosceles triangles b: The central vertex must be = 36. The other two angles must be equal since the triangle is isosceles. Therefore, ( ) 2 = 72. c: 10!!14.5 = 145 square inches a: (6.5, 5) b: 3 8 c: Using the strategy developed in Lesson 7.3.2,!x = 14 " 2 = 12 and!y = 10 " 2 = 8. Then the x-coordinate is (12) = 6.5 and the y-coordinate is (8) = a: x + x + 82 = 180, x = 49 b: 2(71 ) + x = 180, x = a: The region can be rearranged into a rectangle with dimensions 14 and 7 units B b: (14)(7) = 98 square units Selected Answers 55

56 Lesson The reflections are all congruent triangles with equal area. Therefore, the total area is (6)(11.42) = square inches a: 1.04 b: f (t) = (1.04) t c: $199, a: Non-convex b: Convex c: Convex d: Non-convex a: 64 units 2 b: 27.0 units 2 c: 8 3! 13.9 units a: 3 b: 15 c: 4 d: All circles are similar, or use similarity transformations to justify the similarity a: A = 192 cm 2, P = 70 cm b: The length of each side is 5 times the corresponding side in the floor plan. A = 4,800 cm 2 and P = 350 cm. c: The ratio is 5 1 = 5; the ratio of the perimeters equals the zoom factor. d: The ratio of the areas is 25 1 = 25. The ratio of the areas equals the square of the zoom factor (5 2 ). 56 Core Connections Geometry

57 Lesson a: The interior and exterior angles must be supplementary = 160. b: Use = 18 sides or solve the equation 180º(n!2) n = 160 to find n = a: The perimeter of both triangles is units. b: 32.9 and Since the diagonals of a parallelogram bisect each other, they must intersect at the midpoint of BD. Thus, they intersect at (6, 21) A = 100 3! mm meters a: w = ± 17 5! ±!1.84 b: w 2.17 and 1.57 c: No solution E a: 60 b: 82 c: 14 d: a: Equilateral triangle b: Rectangle c: Nonagon d: Rhombus or kite The x-coordinate must lie on the perpendicular bisector of segment AB. Thus, since the midpoint M of segment AB is (6, 0), the x-coordinate of point C must be 6. ΔAMC is a right triangle and the hypotenuse must have a length of 12 units for ΔABC to be equilateral. Therefore, MC = 12 2! 6 2 = 6 3 because of the Pythagorean Theorem. So the y-coordinate of point C could be 6 3 or a: Yes; since BC = BC (Reflexive Property), AB! DC (given), and ABC DCB (given), then ΔABC ΔDCB (SAS ). Therefore, AC = DB ( Δs parts). b: No; the relationships in the figure are true, as long as the two angles remain congruent. See the diagram for problem 8-30 for a similar diagram a: ( 2.5, 0) and (3, 0) B D b: The graph of y =!!(2x 2! x!15) would be the reflection of y = 2x 2! x!15 across the x-axis because each y-value would have its sign changed. Selected Answers 57

58 Lesson a: A = 34 units 2 ; P 25.7 units b: A = 306 units 2 ; P 77 units c: ratio of the perimeters = 3; ratio of the areas = inches or 6.67 feet The area of the hexagon 23.4 ft 2. Adding the rectangles makes the total area 41.4 ft ! 19.05% ; k = 0, 6, 10, 12 are factorable a: Reasoning will vary. For example, it is most likely you will earn more extra credit if the class spins the spinner with the options of 5 and 10 points. b: Reasoning will vary, but now the first spinner is definitely more attractive x 2 = 2x 2 +17x! 30, x = 2.5 or 6; yes, there are two possible answers B 58 Core Connections Geometry

59 Lesson a: 3 4 b: rp c: ar a: cm 2 b: cm a 150 =! a: = 6, so two sides will collapse on the third side. b: Answers vary. One solution is 2, 5, and // Given in diagram A C Given in diagram ABD CDB If lines are //, then alt. int. angles are. Shared side (or Reflexive Property) ΔADB ΔCBD AAS Δs parts a: AAS, ΔABC ΔDCB D b: ASA, ΔABC ΔEDC Selected Answers 59

60 Lesson Area of the entire pentagon square units, so the shaded area 5 3 (172.05)! square units a: x = 14 3, pattern b: x 5.78, Law of Sines. c: No solution, hypotenuse must be longest side. d: 24 units, triangle area formula BC! DC and A E (given) and BCA DCE (vertical angles are ). So ΔABC ΔEDC (AAS ) and AB! ED (!!s "! parts ) a: x + x + 125º + 125º + 90º = 540º, x = 100 b: 6x + 18º = 2x + 12º, x = a: (1.5, 5) b: (3, 7) c: y = 4 3 x + 3 d: = 6.25 = 2.5 units B Lesson (100 25π) square units a: 8 b: 18 c: Could have the combinations (1, 33), (3, 11), (11, 3) or (33, 1) a: A = 42 square units, P 30.5 units b: A = square units, P! units a: x = 26; if lines are parallel and cut by a transversal, then alt. int. angles are equal. b: x = 33, n = 59 ; If lines are parallel and cut by a transversal, then same-side exterior angles are supplementary a: 20 b: units The area of the hexagon is 24 3 square units, so the side length of the square is 24 3! 6.45 units D 60 Core Connections Geometry

61 Lesson a: C = 28 π units, A = 196 π units 2 b: C = 10 π units, A = 25 π units 2 c: d = 100 units, r = 50 units, area = 2500 π units 2 d: A = C 2 /4π units See bold answers in table below. Statements 1. AB! DE and DE is a diameter of!c. 1. Given Reasons 2. AFC and BFC are right angles. 2. Definition of Perpendicular 3. FC = FC 3. Reflexive Property 4. AC = BC 4. Definition of a Circle (radii must be equal) 5. ΔAFC ΔBFC 5. HL 6. AF! FB 6.!!!s!"!!!parts a: (3) b: (1) c: (4) d: (2) a: 3 b: 4 c: 3 and 3 d: 2 and a = 97, b = 15, c = 68, d = D a: (55)(60) + 900! " square feet b: π feet, = $ or approximately $2,388 c: Area is four times as large 24,509.6 square feet; perimeter is twice as long 597 units a: CD = 22, BC = 7, and ED = 6 ; the perimeter is = 48 units b: 54(4) = 216 cm a: 16 3! square units b: 36 square units; more c: 24 3! sq. units; its area is greater than both the square and the equilateral Δ. d: A circle a: 1 8 b: a: 2πr = 24; r = 12! ; A = 144! " square units b: 2πr = 18π; r = 9; A = 81π square units E Selected Answers 61

62 Lesson a: Solutions vary. Possible solution shown at right. b: Answers vary. Assuming there are no hidden cubes, V = 11 units a: 4 25 b: 196:1 c: 9: Since the perimeter is 100, each side is 10. The central angle is = 36. The right triangle has acute angles 18 and 72. Area = units (36! 9" ) 2 # 3.86 units a: (4, 2) b: ( 6, 4) c: ( 1 2,!12) a: An area model is not possible because there are more than two events. A tree diagram follows part (b). b: From the following tree diagram, P(long pants) = = 1 2, or, # successes 6 long pants total # possible = 12 students = 1 2. clothing shoes tennis gender long pants other tennis male 0 shorts dress or skirt other tennis long pants other tennis female shorts other dress or skirt tennis other A c: Using M or F for gender, and L, S, or D for clothing, and T or O for shoes, the union is {MLT, MLO, MST, FLT, FLO, FST, and FDS}. The intersection is {MLT, FLT} 62 Core Connections Geometry

63 Lesson a: square mm b: 400% is 4 times as large. Therefore, its area increases by a factor of 4 2 ; ,778.6 square mm a: See solutions at right. b: 42 units a: 151 b: Yes; a regular 36-gon has interior angles of 170. c: = 4860 F R T a: 2x + 4x! 3+ 7x! 6 + 3x x +10 = 540, x = 31 b: 4x x! 2 = 180, x = or 72.9% decrease a: 2 2 units b: ( 1, 6), (x, y)! (x, "y) c: (8, 5) d: (2, 3) cm D Lesson square units; As a midsegment, DE must be half the length of BC. If the ratio of lengths is 0.5, then the ratio of areas is = Base Area = cm 2 ; Height = 5 cm; SA = cm Yes, by AAS By the Addition Rule, 0.07 = ! P(long and lost), resulting in a probability of % that the food took too long and the rider got lost a: 6 or 6 b: No solution because absolute value cannot be negative. c: x = 3 or ! (6) 2 (14.5) = 522! in. 3 ; 522!in.3 1!! 1!gallon 231!in. 3 " 7.1!gal Selected Answers 63

64 Lesson If she needs the balloon to double in width, then the volume will increase by a factor of 8. That means the balloon requires 24 breaths to blow it up. Since she has already used 3 breaths, she needs 21 more to fill the balloon a: SA = 180π in. 2 ; V = 324π in. 3 b: 324π 27 = 27, in Circumference of each circle = 10π; total distance = 20π 62.8 feet a: x 10.3 b: No solution because the hypotenuse must be the longest side of a right triangle. c: The length of the base of the composite triangle must be 6 3. The smaller triangle has a base length 6 3! 5 " 5.39 ; x a: See diagram at right. b: V = 16 cubic units; SA = 52 square units The line should be solid, and the shading should be to the right of the line See the area model below. A tree diagram would have worked as well = 45 7 red oak 9 3 white oak 4 9 maple 2 9 granite tile D F R T 64 Core Connections Geometry

65 Lesson b: 2 c: 24 and 96; ratio = 4; It is the square of the linear scale factor. d: 6 and 48; ratio = 8 = 2 3. It is the cube of the linear scale factor a: Height of the tank = 6 3! 10.4 in., so V = 7!!13!!6 3 = 546 3! inches3 b: c: 25 fish fish in3! 25 fish in3! 12 inch 1 foot! 1in 3 or about fish per cubic inch. 12 inch 1 foot! 12 inch fish 1 foot " 1ft a: False (isosceles trapezoid) b: True c: True d: False (parallelogram) (12x + 7) = 30x! 4, so x = a: x = 2 b: x = 5,!! 1 2 c: x = 2 d: x = 2, a: 0.85 b: f (t) = 27000(0.85) t c: $11, a: It is possible D b: Not possible. Same-side interior angles should add up to 180 c: Not possible. One pair of alternate interior angles are equal, but the other is not for the same pair of lines cut by a transversal; or, the vertical angles are not equal. Selected Answers 65

66 Lesson a: A rhombus b: They are perpendicular and intersect each other at their midpoints. Or, the diagonals bisect the angles of the quadrilateral. c: units 2 d: units a: See diagram at right. b: 3000 ft 3 c: 4.8 ft 3 d: 0.2 butterflies per cubic foot because 625butterflies 3000ft x! 5 = 2x + 9 ; x = a: Not congruent since side ratio 1 b: ΔABC ΔDBC (HL ) c: ΔABC ΔMLK (SSS ) y = 6x! See the answers in the Checkpoint Materials section B! 0.21 butterflies ft 3. 15' 20' 10' 66 Core Connections Geometry

67 Lesson b: It got longer. c: Although the change is minute, the line segment got a bit longer. d: It is possible, although not using Karen s incremental strategy. One way: Construct a hypotenuse for a right isosceles triangle with leg lengths 1 cm Possible response: Construct a circle of any radius. Then choose a point on the circle and mark off two radius lengths in each direction along the circle circumference. Connect the original point to each of the other two points Area of whole circle = 16! m 2 ; Area for dog to roam = (16! ) = 32! 3 " m Height = 9 3 m, so A = ! " ft 3 (20+14)(9 3) 2 = 153 3! 265 m Equilateral triangle: sides = 120 ; regular heptagon: (7!2)!!180 7 " 128.6, so the measure of the interior angle of a regular heptagon has greater measure See graph at right. x-intercept: (4, 0), y-intercept: (0, 6) C y x Selected Answers 67

68 Lesson a: See diagram below right. b: 24 cubic units c: The volume of the new solid must be 1 8 reduced volume must be 3 cubic units. the original, so the mat plan ΔABC ΔEDF (given), so B D (!!!s!"!!!parts ), so DG! BG (if base angles of a triangle are congruent, then the two sides opposite those angles are also congruent) and ΔDBG is isosceles (definition of an isosceles triangle) a: 74.2 m 2 b: 36.2 meters c: 6 goats 74.2 m goats 2! 0.1 2, or approximately 1 10 m goat per square meter Using equations: n + q = 14 and 0.05n q = 2.90, so n = 3 and q = a: It has 8 sides. b:! ft DA! CB and m DAB m CBA (given), AB! BA (reflexive property), so!dab!!!cba (SAS ). Then C D (!!!s!"!!!parts ) f (x) = 7.68(2.5) x B 68 Core Connections Geometry

69 Lesson a: 14! " cm 3 b: BA 19.3 ft 2, so V (19.3)(7) ft She should construct an arc centered at P with radius k so that it intersects n and m each once (call these intersections R and S). She should then construct two more circles with radius k, centered at R and S. The fourth vertex lies where these two circles intersect = 180,000 y square miles a: See graph at right. b: f (x) = 10(2.3) x x c: y = 42, 000(0.75) 5, 9967 d: 60 = 25(b) 10,!b = 1.09, 9% increase a: b: C Lesson a: V = (2)(5)(6)! "(0.5 2 )(6) # 55.3 cm 3 b: Answers vary. One possibility: It could represent a pencil sharpener a: 70 b: 50 c: 2x She is constructing an angle bisector Height = 6 3, area = 15(6 3) (144!) = ! " in.2 ; perimeter = (24!) = ! " 66 in a: 17,!17,17,!17,17 ; geometric b: 32,11, 1 2,!4 3 4,!7 3 8 ; neither c: 81, 81, 81, 81, 81; arithmetic and geometric They intersect only once at (3, 5) D Selected Answers 69

70 Lesson a: 64 b: 128 c: 64 d: 180 e: 128 f: Central angle = 3.6 ; A square units a: 5m +1 = 3m + 9, m = 4 b: 2(x + 4! ) = 3x! 9!, x = 17 c: (p! 2) = p 2, p = 10 d: 18t = 360!, t = a: D(0, 4) and E(4, 7) b: DE = 5 units, so AC should be 10 units long. c: = 10 units b: 108, inscribed angle c: 72, central angle d: 216, yes a: 9 cubic units b: 10 cubic units D c: Possible response: There are no floating blocks and that there are no hidden blocks behind the ones visible. d: Either a left or back view would reveal any hidden blocks. Lesson a: 3 b: 6 c: 2 d: 1 e: 4 f: a: They are similar. CD! is a dilation of AB! from P. b: They have the same measure because they have the same central angle. However, CD! is longer; it is similar to a part of a circle with shorter diameter. c: (28! ) " 14.7 units OY! KY! EY! PY (all radii are congruent),!pyo "!EKY (arc measures are equal), so!poy "!EKY (SAS! ). Therefore, PO! EK because! "s #! parts It must be a rhombus and it could be a square ! (2) 2 (4) 45 ( 360 ) " 6.28 ft a: Yes (AA ~), ΔABC ~ ΔLKH b: Not enough information C c: Yes (AA ~), ΔABC ~ ΔEDC 70 Core Connections Geometry

71 Lesson a: 50 b: 50 c: 67 d: 126 e: 54 f: a: x 31.9, y 10.5 in. b: x 3.7 m c: x 34.7, y = 250! 15.8 ft a: 4 times b: 360 c: 360! 5 = area of the regular pentagon! 61.9 ft 2 ; total SA! 2(61.9) + (5)(6)(12)! ft 2 ; volume! (61.9)(12)! ft a: It is a square. Demonstrate that each side is the same length and that two adjacent sides are perpendicular (slopes are opposite reciprocals). b: C! is at ( 5, 8) and D!! is at ( 7, 4) The angles, from smallest to largest, measure 64, 90, 116, 130, and 140, so the probability is C Lesson MA = = 39 feet, MB = 6 feet, so AB = ER and ER = 1485! 38.5 feet a: The arc length is !2"(3) = " 2, so the ratio 2 3 =! 6. b: The arc length is 5! 6, so the ratio is 6 5 =! 6. c: Possible response: Because all circles are similar, circle D is similar to circle E. Therefore sector D is similar to sector E (the angles are the same and the radii and arc lengths are proportional by the same ratio of similarity). Therefore we can write a arc length D arc length E proportionality equation, radius D = radius E. In fact all sectors with 30º central arc length angles will be similar, so the ratio of radius, that is, the radian, will be the same for all 30º sectors (9)(12) 1 2 (20)h, h! 5.4 " The ratio of the volumes must be 1 ( 5 ) 3, so the volume must be 1 ( 5 ) 3 ( 500! ) = 4! cm a: 1.05 b: $130,588 c: f (t) = (1.05) t a: A sq. cm b: A sq. cm c: A sq. cm E Selected Answers 71

72 Lesson a: No, there may be red marbles that she has not selected in her draws. b: No, it is less likely that there are red marbles, but no number of trials will ever assure that there are no red ones. c: This is not possible, no number of draws will assure this a: 1 6 ; 1 2 (60) = 50 times b: 5 6 c: Answers vary, but two squares should have an even number, no numbers can be 3, and all numbers must be less than a: 13 b: 6.5 c: 67.4º d: 134.8º a: Have Ken s analyzed for $30, and use the ratios of similarity to calculate the data for Erica s nugget themselves. b: Since r = 5, the ratio of the areas is 25. Thus, 20!25 = 500 cm 2. c: Since r = 5, the ratio of the volumes is 5 3 = 125. Thus, Erica s nugget weighs 125!5.6 = 700 g, which is about 25 ounces x + 3x + 4x + 5x, x! a: True b: False c: False d: True B 72 Core Connections Geometry

73 Lesson a: x = y b: y = 2x or x = 1 2 y c: 3y = 5x d: x + y = a: Yes, because of the Triangle Sum Theorem, 180! 64! 26 = 90. b: Yes, because = a: See diagram at right. b: 1.4 ft a: See diagram at right. b: = 25 36! 69.4% c: 5 6! 5 6 = " 69.4% (49! ) " square cm d: e: 1! 1 4! 2 3 = 12 1, 12 1 (360 ) = 30 blue red 0.2 ft 1.5 ft 0.25 ft 0.2 ft 0.25 ft 1 ft 1 ft a: Both are 4!. Note that a radian measure is a ratio and does not have units. x (2! r) b: If x is the unknown central angle of the circle, then the equation, 360 r solved for x. x = 60º B a: P(on-campus given Engr) = ! 13.0% b: P(on-campus) = 0.6 Selected Answers 73 =! 3, can be c: No. The probability of living on campus given that the student is an engineer is much smaller than the probability of living on campus Region A is 1 4 of the circle, so it should result 1 4 (80) = 20 times. Regions B and C have equal weight (which can be confirmed with arc measures), so they should each result (80 20) 2 = 30 times a: = 40 b: mad! = 2(97 " ) = 194 " ; m!c = 0.5(1.94) = 97 c: mab! = 125º and the length of AB! = 125º 360º (16! ) " 17.5 "; area = 125º 360º (64! ) " 69.8 in Methods vary, a variety of relationships such as Parallel Line Angle Conjectures, the Exterior Angle Theorem, or the Triangle Sum Theorem can be used. x = 109, y = 71, z = a: x = 34 b: x = 4 3 c: x =!5 d: x = She is incorrect, which can be tested by substituting both answers into the equation; w =!6 or A

74 Lesson a: = 46.3% b: P(laptop) P(laptop given business trip) so they are associated a: See table below. Entries not in bold are given in problem statement, and entries in bold are computed from given information. Daily Not Weekly local 25% 12% 37% Not weekly local 40% 23% 63% D a i l y b: 25% + 12% + 40% = 77% c: ! 84.4% 65% 35% 100% =! (2.5 2 )h, h = ! " 5.49 in ! ; Since the angle is doubled, the arc length must be doubled, and in turn, the ratio of arc length to radius, or radian measure, must also double and thus be! a: See views at right. Depending on how they are oriented, the top and bottom views are the same. (Or they could be reflections of each other.) b: V = 7 units 3, SA = 28 units 2 ; Methods vary. F R T c: Since this solid has no holes, the surface area can be calculated by adding the areas of each of the views b: AB AC e: AB AC c: BC AB f: BC AC d: BC AC a: If x represents the length of chord AC, then x 2 = ! 2(10)(10) cos 80º ; x 12.9 b: Core Connections Geometry

75 Lesson a: 24 b: The decision chart tells how many branches there are at each stage. c: ! = a: = 20% b: 45 9 = 20% c: Yes a: See table at right. b: 1 9 c: ! 66.7% Edge = 1200/12 = 100 cm, SA = 6(100 2 ) = 60, 000 cm 2 ; V = = 1, 000, 000 cm Area of shaded region! 93.53" 84.82! 8.71 in C SQUARES yellow red blue red CIRCLES yellow 1 Lesson a: 7! = 720 b: P 3 = (12!3)! 12! = 12! 9! = 12 "11"10 = No, they should not charge a higher premium. P(ticket given red) = ! and P(ticket) = 20, ! Since they are approximately the same, they are most likely independent a: 30% b: 42% c: P(Green Fang)! P(alarm) " P(Green Fang and alarm), 0.68!0.28 " 0.22, so they are weakly associated ,184, 000!17, 576, 000 = 140, 608, 000 ; a: cylinder b: triangle based prism c: cone Selected Answers 75

76 Lesson != 362, a: 1 b: 8 things taken 8 at a time. 8 P 8 should equal 8!. 8 P 8 = 8! 0!= 0 would be dividing by 0. c: 3! 3 = 2!, 2! 2 = 1!, 1! 1 = 0! a: 10 P 8 = 10! 2! = 1, 814, 400 b: 10 C 8 = 10 P 2 8! = 10! 8!2! = 45 c: 6C 1 = 6! 1!5! = 6 (8!8)! = 8! 0!. If, you Because they are independent, P(coffee and dairy) = P(coffee) P(dairy) = ( )( )! 22.2%. P(no caffeine and no dairy) = P(no caffeine) P(no dairy) = ( 63!42 63 )( 63!21 63 ) " 22.2%. Avoid the common mistake of computing P(no caffeine and no dairy) = 1 P(coffee and dairy) because {coffee and dairy} includes all the drinks that have coffee or have dairy but do not have both coffee and dairy a: 16 P 6 = 16!15!14!13!12!11 = 5, 765, 760 b: 1! 15 P 5 = 1!15!14!13!12!11 = 360, ,360 c: 5,765,760 = 6.25% d: 15 C 5 16 C = = 37.5% a: 20 b: 30 c: a: 5!= 5! 4! 3!2!1 = 120 b: 5! 2! = 5!4!3!2!1 2!1 = 60 c: 5! 2!2! = 5!4!3!2!1 2!1!2!1 = 30 d: Because you cannot tell the repeated letters apart, there are fewer arrangements when there are repeated letters. 76 Core Connections Geometry

77 Lesson C 3 = 20, 708, ! 7!2!1! 4 = 168 choices a: 22 P 3 = 9240 b: This is really a permutation lock. In this case the common use of the word combination conflicts with the mathematical meaning. c: 22 C 3 = 1540, but this does not make sense for a mechanical lock because it would imply dialing the numbers in any order to open the lock. d: 22!21!21 = a: 162º b: 16 sides c:! cm a: 12; (12 1), (12 2), (12 3) b: n(n 1)(n 2)(n 3)(n 4)(n 5) c: n(n 1)(n 2)(n 3)(n 4)(n 5) a: There is one way to choose all five. 5! 5!0! = 1. In order to have the formula give a reasonable result for all situations, it is necessary to define 0! as equal to 1. b: There is one way to choose nothing. 5! 0!5! = 1 c: This is a permutation in which some of the items occur multiple times, like in an anagram. 8! 2!5! = 168 arrangements of cars. Selected Answers 77

78 Lesson C 3 = 161, The first because 3!> 2! a: (n! 3)(n! 4)(n! 5)(n! 6)(n! 7) b: n + 2, n +1, n!1, n! 2 c: n(n!1)(n! 2) d: (n + 2)(n +1)(n!1) a: See diagram at right. Entries not bolded are given in the problem statement, while bold entries are computed. b: P(Senior!Ocean View) = = 25% c: They are associated because P(senior)! P(Ocean View) " P(senior and Ocean View), this is (0.10)(0.24)! (0.06). Not OceanView Ocean View Senior (0.60)(0.10) = Not senior (0.20)(0.90) = or a 17% decrease a: See diagram at right. b: h 2 = 17 2!15 2 = x 2, x = 8, A(base) = 0.5(8)915)! 60, 600 = 60h, h = 10 cm c: TSA = 2(60) + 17(10) + 8(10) + 15(10) = 520 sq. cm d: 90º; sin a = 17 8, a = 28º; sin b = ; b = 62º C 5, 12 C 4, 12 C 3, 12 C 2, 12 C 1, 12 C 0 = cm 15 cm This is permutation with some of the items to be arranged occurring multiple timers, like an anagram. 3!2! 10! = 302, 400 ways to layer the dip If t = hours, t = 9 ; 28 hours a: Similar b: Similar a: b: FRONT FRONT RIGHT RIGHT SA: A(!) = 1 2 (8)(4 3) 27.71; A(face) = (8)(10) = 80; TSA = 2(27.71) + 3(80) sq. cm. V = 27.71(10) cu. cm. 78 Core Connections Geometry

79 Lesson a: Yes; the sum of the interior angles only depends on the number of sides of the polygon. b: No; only regular hexagons have a guarantee of having an angle with the measure 120. c: Yes; that is the definition of hexagon a: 7!= 5040 b: 1!5!!1 = Lateral surface area = (circumference of the base)(height) = 8! "15 = 120! # 377 cm a: 2x = 180!106, x = 37 b: x + 67 = 180, x = 113!, 5y + 3y!16 = 180, y = 24.5 c: sin 73 9 = sin 57 x = sin 50 y, x! 7.9, y! 7.2 d: 4x! 2 + 2(8x! 9) = 180, x = a: A = 144 square units, P = 84 units b: A = 16 square units, P = 28 units a: A sphere B b: A cylinder with a cone on top and bottom. c: A double cone: two cones attach at the vertices a: 4 b: 6 c: n! (n!4)! (10)(12)h = 840, so h = = 7 mm a: b is larger, even though we are not told that b is a central angle. b: The missing angle = 180! 62! 70 = 48 and since the angle opposite side a is bigger, a must be larger than b. c: a = 9 3! 15.6 units 2 and b = 16 units 2, so b is larger than a a: x 2 + y 2 = r 2 b: sin! = y r ; y c: cos! = x r ; x If the circle s center is C and if the midpoint of AB is D, then ΔADC is a triangle. Then the radius, AC, is 10 units long and the area of the circle is 100! " units C Selected Answers 79

80 Lesson inches a: 23 P 3 = 10, 626 b: 23 P 3 = 10, 626 c: 1!22!22 = 484 d: 44!22!21 = The areas are all equal because the triangles have the same base and height a: x = 12.9 b: x = 2 c: x = 0 c: x = 23 9! PA = PA (Reflexive Property), m PBA = m PCA = 90º (tangents are to radii drawn to the point of tangency), PB = PC (radii of a circle must be equal), so ΔPAB ΔPAC (HL). Therefore AB = AC (!!!s!"!!!parts ) a: 10 b: tan! = 8 6,!! " ±6, 18, ± D Lesson (Day 1) a: Yes. One way is to cut off a corner so the cross-section is a triangle. b: A tetrahedron (also called a triangle-based pyramid) V = 1 3 (62 )(4) = 48 units 3 ; slant height = = 5; SA = 4( ) + 62 = 96 units a: 900 C 12! 5.48 "10 26 b: 899 C 11! 7.30 "1024 c:! 1.3% a: 8 faces, 12 edges, and 6 vertices b: A square a: x = 12, y = 7.5 b: 4 3 c: 48 square units A = 16! square units; C = 8! units B 80 Core Connections Geometry

81 Lesson (Day 2) region B, since area! units 2 and perimeter 24.3 units a:! (1.5) 2 (4.5) " 31.8 in. 3 b: Volume of the pot is! (7) 2 (10) " in. 3. Therefore, Aimee would need = 48.4 or 49 cans of soup to fill the pot. c: 2! (1.5)(4.5) " 42.4 in m ECB = m EAD (given), AE! CE (definition of midpoint), DEA = BEC (vertical angles are congruent), ΔAED ΔCEB (AAS ), so AD! CB (!!!s!"!!!parts ) a: 6 P 4 = 6!5! 4! 3 = 360 b: 1!5! 4! 3+ 5!1! 4! 3+!5! 4!1! 3+ 5! 4! 3!1 = 4! 5 P 4 = a: ! = = 46.9% b: 1! = = 53.1% c: 1! = = 75% d: P(A given under 20) = = 60.8% a: 18x +174 = 540, x = b: x2 + (x +17) 2 = 25 2, x = E c: 2(x + 4 ) = 134, x = 63 d: x x +15 = 180, x = 39 Selected Answers 81

82 Lesson a: BA = 1 2 (7)(24) = 84 in.2, V = (84)(12) = 1008 in. 3, SA = (2)(84) + (12)( ) = 840 in. 2 b: BA = 25! 2, V = 1 3 (25! )(12) = 100! m3, lateral SA =! (5)(13) = 65! m 2, total SA = 25! + 65! = 90! " m a: 8 C 3 = 56 b: There are 6 choices left for the third filling. c: 7 C 2 8 C 3 = = 37.5% Only one right triangle can be built: 3-4-5; 6!6!6 3!2!1 = You cannot use combinations because each combination is not equally likely. For example, rolling a combination of 1, 1, 2 on the dice (where order does not matter) is three times more likely than 1, 1, a: Vertically through the vertex of the cone. b: A circle c: A sphere a:!abc! FED (AA~) b:!abc ~!MKL (SSS~) c: Not similar because the zoom factors for corresponding sides are not equal See graph at right a: Corresponding angles,! A b: Alternate interior angles,! c: Straight angle, supplementary d: Alternate exterior angles, neither because lines intersect. 82 Core Connections Geometry

83 Lesson r = 4 cm, SA = 4! (4 2 ) = 64! " cm 2, V = 4 3! (4)3 = 256 3! " cm C 3! 10 C 3 = a: 12 C C 3 = 715 b: If raspberry and custard are known filling, then there are two fewer fillings to choose from, so 10 C C = ! 7.7% small cone: 2 5 = r 6, r = 2.4"; V = 1 3! (2.4)2 (2) " in. 3 ; large cone: V = 1 3! (6)2 (5) " in. 3 ; new volume = ! 2(12.06) " in a: An icosahedron has 20 faces, so the surface area is (20)(45) = 900 mm b: Since it has 12 faces, = 9 cm 2. c: The area of each face is 1 2 (6)(3 3) = 9 3! 15.6 in.2, so total SA = 4(15.6)! 62.4 in ! square units a: ( 4 3,!10) b: (!1,!9) Selected Answers 83

84 Lesson a: Earth s circumference: 8000! = 25,132.7 miles. Therefore the distance to the moon is 230, ,132.7! 9.5 times greater. b: cos = 238,900 x, x! 91, 253,182.4 miles C! 102.6, so mab!! (102.6)! 9.1 mm a: 1 3 (92 )(12) = 324 cm 3 b: 12 cm a: 52 C 2 = 1326 b: 16 C = ! 9.0% or using permutations 16 P 2 52 P 2 = 16!15 52!51 " 9.0% c: 16C 2 52C2 d: 12 C 2 52 C 2 = ! 5.0% or using permutations 12 P 2 52 P 2 = 12!11 52!51 " 5.0% e: 4 C 1! 12 C = " 3.6% or using permutations, 2! 4!12 52!51 " 3.6% a: ! 61% b: = 50% Answers vary. Sample responses: The rings in a tree, the ripples created when a stone is tossed in a pond, the rings of a dartboard, etc D 84 Core Connections Geometry

85 Lesson radius of slice = 3! " in., x 2.5 2! " 2.3 inches The surface area of the moon! 4" (1080) 2! 14, 657, which makes it larger than Africa and smaller than Asia C1!6C1!2C1 = 12 12C3 55 " 21.8%, or permutations and the Fundamental Principle of Counting could be used: there are 3 P 3 = 6 ways to arrange the colors, and 12 P 3 = 12!11!10 ways to arrange twelve pens, so, 6! 6!4!2 12!11!10 = 21.8% Central angle = 36, distance from center to midpoint of side = units, A = 1 2 (20)(307.77)! square units V = 324!12 = 312 cm !ART!!PIT by AA~ a: V = 2100 units 3, SA = units D b: V = 1000! cm 3, SA = 1100! " cm 2 c: V = 60 in. 3, SA = 144 in. 2 Selected Answers 85

86 Lesson a: x = 270 b: x = 132, y! 15.7 c: 3(x + 2) = 6x, x = a: a = 44, b = 28, c = 56 b: Some students may just report that c is smaller. But some may notice that the vertical angles (72 ) are each the average of the two arcs they intercept. This will be revisited in problem Since P(academic) P(Arts) = P(academic and Arts) the events are independent. There is no association between winning an academic award and a Fine Arts award. Another possible method is P(Arts given academic) = P(arts), or that P(academic given Arts) = P(academic) Base area = 36 units 2, slant height = 109! units, lateral SA = ! units 2, total SA! ! units a: 436,000 miles b: The sun s radius is almost double the distance between the Earth and the moon. That means that if the sun were placed next to the Earth, its center would be farther away than the moon! c: 1,295, V(16)(16)(16) = 4096 units 3 ; SA = (6)(16)(160 = 1536 units C a: x = 117, y = 88! 9.4 b: r = 5 sin 25! 11.8, z = 310 c: 9(9 + a) = 8(21), a = V(prism) = (34)(84)(99) = 282, 744 units 3 ; V(cylinder) =! (38) 2 (71) " 322, units 3, so the cylinder has more volume C 3 + 2! 6 C C 1 = (15) + 6 = x = 2(62 ), x = f (x) = 4( 3 2 )x The solution is shown with dashed lines in the diagram above D 86 Core Connections Geometry

87 Lesson a: x 2 + y 2 = 9 b: a: V = 1 3! (32 )(10) = 30! " 94.2 units 3 b: One method: BA = (21)(18) (12)(12) = 234 units 2, V = (234)(10) = 2340 units Think of this as an anagram ( 4!2! 6! = 15 ) a: b: f (t) = 8500(1.005) t c: $11, a: x = (x + 2) 2 ; x = 15 b: tan!1 ( 15 8 ) " 28.1, 180! 90! 28.1 " 61.9 ; Sample tools: Trigonometry and the Triangle Sum Theorem a: 124 b: 25! units 2 c: 12.3 units C, by SAS Selected Answers 87

88 Lesson (Day 1) a: The slant height of the cone is 9.22 m, LA(cone) 6π(9.22) m 2, and LA(cylinder) = 12π(11) = 132π m 2, so total surface area is = m 2 b: V(cylinder) = 36π(11) = 396π m 3 and V(cone) = 1 3 (36! )(7) = 84! " m3, so total the volume is = m a: 36 b: b = c = 108, d = 72º a: E(1, 3) and F(7, 3) ; AB = 9, DC = 3, EF = 6 ; EF seems to be the average of AB and CD. b: Yes; EF = 4, while AB = 6 and CD = 2 c: Sample response: The midsegment of a trapezoid is parallel to the bases and has a length that is the average of the lengths of the bases a: (2,!3), r = 5 b: (!1,!3), r = a: sin 27 = x 18, x 8.17 b: sin102 7 = sin 62 x, x 6.32 c: tan x = 6 4, x a: Vertical angles are equal, 2x + 9 = 4x! 2, x = D b: The sum of the angles of a quadrilateral is 360, (3x + 8) (2x!1) = 360, x = 41 c: When lines are parallel, same-side exterior angles are supplementary, so 7x! 3 + 4x +12 = 180 and x = Core Connections Geometry

89 Lesson (Day 2) a: BA m 2, LA = (8)(4)(16) = 512 m 2, total SA 2(77.25) m 2 b: Slant height = 12 ft, LA = 4( 1 2 )(10)(12) = 240 ft2, BA = (10)(10) = 100 ft 2, total SA = = 340 ft a: u =!1 b: x = 5,!! 8 3 c: k = 1!or!9 d: p =!4.5!or! a: Since the hypotenuse is 1, sin! = y 1, and y = sin!. Also, cos! = 1 x, so x = cos!. b: It must be 1 because of the Pythagorean Theorem. c: Yes, this appears to be true for all angles a: (x! 4) 2 + (y! 2) 2 = 9 b: Pick any two points, such as those that are parallel to the axes on the circle and then find the distance between the points using the Pythagorean Theorem a: 2!5!5 = 50 b: 2! 4! 3 = a: k 2 = (8)(18) = 144, k = 12 b: k = A Selected Answers 89

90 Lesson Since 2πr = 40 feet, then r = 20! V = 4 3! ( 20! )3 " ft feet; SA 4! ( 20! )2 " square feet; a: x = 6 b: x = 4 or 4 c: x = 4 d: x = 30 e: x = 3 or 3 f: x = 3 or Answers vary. Typical cross-sections: regular hexagon, circle, rectangle, etc a: Yes; the graph includes the circle and all of the points inside the circle. b: No; the graph only includes the points outside the circle. The circle itself would be dashed This is an anagram of five Hs and five Ts. 10! 5!5! = 252. It is not a combination because order matters a: The point is not on the circle. This can be shown using the fact that all of the points on the circle are 3 units away from the origin and then finding the distance from the origin to the point (1, 5) with the Pythagorean Theorem, ! B b: x = 2 or 2 c: Possible answers will satisfy the equation x 2 + y 2 = Core Connections Geometry

91 Lesson V = 820( 1 2 )3 = cm See solution graph at right. a: C: (0, 0); r = 4.5 b: C: (0, 0); r = 75! 8.7 c: C: (3, 0); r = 1 d: C: (2, 1); r = 19! a: x = 33, y = 120 b: a! 36.9, b = 4 c: z = 12, w = 5 d: x = a: FG = 3.5 cm, BC = 14 cm b: 16(3) 4(3) = 36 cm base radius = 14 in.; V = 1 3 (196! )(18) = 1176! 3695 in.3 ; SA =! rl =! (14)( 520) " 1003 in a: 4 C 3 12C3 = = 1.8% B b: 5C 2! 4 C 1 12C3 c: 5C C C 3 12C3 = " 18.2 = ! 6.8% Selected Answers 91

92 Lesson ; One method: The base angles of ΔPQR must add up to 40 so that the sum of all three angles is 180. Then add the 40 and 35 of QPS and QRS, respectively, and the sum of the base angles of ΔPRS must be 115. Thus, m Q must be = A! 1, 459, square feet a: No; the triangle is equilateral, so all angles must be 60. b: Yes; = See graph at right. x-intercepts: (4, 0) and ( 4, 0), y-intercepts: (0, 2) and (0, 8) a: This has one solution, because 4( 23 4! 3) = 11. b: This has no real solution because x 2 must be positive or zero. c: This has two solutions because x = ± 6. d: This has no solution because the absolute value must be positive or zero See graph at right C Core Connections Geometry

93 Lesson a: a = 120, b = 108, so a is greater. b: Not enough information is given since it is not known if the lines are parallel. c: Third side is approximately 8.9 units, so b is opposite the greater side and must be greater than a. d: a is three more than b, so a must be greater. e: a = 7 tan 23! 2.97 and b = 2 cos 49! 3.05, so b is greater than a x = 8 and y = See graph at right. a: y-intercept: (0, 8); x-intercepts: (3, 0) and ( 1, 0) b: (1, 8) c: f (100) =!19, 594 and f (!15) =! π sq. units (5!)(3!) = SA = 76 units 2 ; V = 40 units B Selected Answers 93

94 Lesson Side length = 4, so height of triangle is 2 3. Thus, the y-coordinate of point C could be 2 ± 2 3 ; (5,!7.46) or at (5,!!1.46) length of diagonal = 4sin54º , or 32! 32 cos108 " (using the Law of Cosines) height of shaded triangle = ! 2 2 " units, area of triangle =! 1 2 (4)(6.155)! sq. units The graph should include a circle with radius 5, center (0, 0), and a line with slope 1 and y-intercept (0, 1). (3, 4) and ( 4, 3) Jamila s product does not equal zero. She cannot assume that if the product of two quantities is 8, then one of the quantities must be 8; Correct solution: x =!6!or! a: 3C 1! 10 C 3 13C C = " 50.3%, or 4 P 1 3 P 1! 10 P 3 13 P 4 b: 11 13C4 = ! 1.5% or 4 P 4 11 = 24! 11 13P " 1.5% = 4! 3!10!9!8 13!12!11!10 = 50.3% Base length units, perimeter units, area square units a: m PCQ = 46º, tan 46 = CP 5, CP = CQ 4.83, sin 46 = 5 CR, CR 6.95, so x b: x 2 = ! 2(7)(7) cos102, x! a: It is a prism with dimensions units. b: SA = 52 units 2 ; V = 24 units 3 c: SA = 52(3 2 ) = 468 units 2 ; V = (24)(3 3 ) = 648 units a: y =! 6 5 x + 4 b: y = 1 2 x! See graph at right A Core Connections Geometry

95 Lesson a: 60 sq. units b: Side length units, so area (7.282)(7)! sq. units a: It was 80 feet above ground because y = 80 when x = 0. b:!16(3) (3) + 80 = 128 feet;!16( 1 2 )2 + 64( 1 2 ) + 80 = 108 feet c:!16x x + 80 = 0 ; x = 5 seconds They all are triangles, so they are all similar to each other. In addition, the triangles in (a) and (c) are congruent because corresponding sides have equal length a: See views at right. b: 11 cubic units c: The volume of the new solid must be 5 3 = 125 times the original, so the increased volume must be 11(5 3 ) = 1375 cubic units. Front Right Top a: C: (!5, 0), r = 10 b: C: (3,1), r = See answers in table below and graph at right. y x y x B Selected Answers 95