How To Understand And Understand Geometry


 Alexis Lawson
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1 Geometry Honors: Circles, Coordinates, and Construction Semester 2, Unit 4: ctivity 24 esources: Springoard Geometry Unit Overview In this unit, students will study formal definitions of basic figures, the axiomatic system of geometry and the basics of logical reasoning. hey will also write equations of parallel and perpendicular lines. Online esources: Springboard Geometry Unit 4 Vocabulary angent Lines adius Diameter Chords rc Conjecture Equidistant Student Focus Main Ideas for success in lessons 241, 242, and 243: Describe relationships among tangents and radii of a circle. Use arcs, chords, and diameters of a circle to solve problems. Describe relationships among diameters and chords of a circle. rove and apply theorems about chords of a circle. rove that tangent segments to a circle from a point outside the circle are congruent. Use tangent segments to solve problems. Example Lesson 241: Given that O = 5, X = 12, and OX = 13.6, is XY tangent to circle O at? Explain your reasoning. Yes, XY is perpendicular to the radius, O, because 13 2 = age 1 of 30
2 Example Lesson 242: he diameter of a circle with center bisects at point X. oints and lie on the circle. Classify the triangle formed by points, X, and. ight riangle Example Lesson 243: In the diagram, NM and QN are tangent to a circle, the radius of circle is 5cm, and MN = 12 cm. What is QN? 12 cm What is N? 13 cm age 2 of 30
3 Geometry Honors: Circles, Coordinates, and Construction Semester 2, Unit 4: ctivity 25 esources: Springoard Geometry Unit Overview In this unit, students will study formal definitions of basic figures, the axiomatic system of geometry and the basics of logical reasoning. hey will also write equations of parallel and perpendicular lines. Online esources: Springboard Geometry Unit 4 Vocabulary angent Lines adius Diameter Chord rc Conjecture Equidistant isecting ay Minor rc Major rc Central ngle Inscribed ngle Interior Half Vertical ngle adii angents Student Focus Main Ideas for success in lessons 251, 252, 253, and 254: Understand how to measure an arc of a circle. Use relationships among arcs and central angle to solve problems. Describe the relationship among inscribed angles, central angles, and arcs. Use inscribed angles to solve problems. Describe a relationship among the angles formed by intersecting chords in a circle. Use angles formed by chords to solve problems. Describe relationships among the angles formed by tangents to a circle or secants to a circle. Use angles formed by tangents or secants to solve problems. age 3 of 30
4 Example Lesson 251: Use the circle shown below to determine the following. What is the measure of the central angle? 96⁰ Write and solve an equation for x. 96 = 8(x); x = 12 What is the measure of the angle that bisects the minor arc? 48⁰ Write the expression that can be used to determine the measure of the major arc. 360⁰  96⁰ age 4 of 30
5 Example Lesson 252: Use the diagram (not drawn to scale) below to answer the following items. a) Which angle(s), if any, are right angles? D b) What is the sum of m D and m DC? Explain. 180⁰. Opposite angles of a quadrilateral inscribed in a circle are supplementary. c) Identify the angle with a measure of 40⁰. D age 5 of 30
6 Example Lesson 253: Write an expression to determine the value of x. x = 1 ( ) 2 Example Lesson 254: Determine m 2, if m 1 = 34⁰ m 2 = 73 age 6 of 30
7 Geometry Honors: Circles, Coordinates, and Construction Semester 2, Unit 4: ctivity 27 esources: Springoard Geometry Unit Overview In this unit, students will study formal definitions of basic figures, the axiomatic system of geometry and the basics of logical reasoning. hey will also write equations of parallel and perpendicular lines. Online esources: Springboard Geometry Unit 4 Vocabulary angent Lines adius Diameter Chord rc Conjecture Equidistant isecting ay Minor rc Major rc Central ngle Inscribed ngle Interior Half Vertical ngle adii angent Completing the Square Student Focus Main Ideas for success in lessons 271 and 272: Derive the general equation of a circle given the center and radius. Write the equation of a circle given three points on the circle. Find the center and radius of a circle given its equation. Complete the square to write the equation of a circle in the form = age 7 of 30
8 Example Lesson 271: Write the equation of a circle given the center and radius. a) Center: (7,2); radius = b) Center: (4,2); radius = 9 Example Lesson 272: Determine if the equation is representative of a circle. If so, determine the coordinates of the center of the circle. Yes; age 8 of 30
9 Geometry Honors: Circles, Coordinates, and Construction Semester 2, Unit 4: ctivity 29 esources: Springoard Geometry Unit Overview In this unit, students will study formal definitions of basic figures, the axiomatic system of geometry and the basics of logical reasoning. hey will also write equations of parallel and perpendicular lines. Online esources: Springboard Geometry Unit 4 Vocabulary angent Lines adius Diameter Chord rc Conjecture Equidistant isecting ay Minor rc Major rc Central ngle Inscribed ngle Interior Half Vertical ngle adii angent Completing the Square Student Focus Main Ideas for success in lessons 291, 292, and 293: Use construction to copy a segment or an angle. Use construction to bisect a segment or an angle. Construct parallel and perpendicular lines. Use constructions to make conjectures about geometric relationships. Construct inscribed and circumscribed circles. Construct tangents to a circle. age 9 of 30
10 Example Lesson 291: Describe a way to divide WO into four congruent segments. Construct the perpendicular bisect of WO. hen construct the perpendicular bisector of each of the two segments that are formed. Example Lesson 292: Construct and label a right triangle with legs of lengths HI and JK. (Hint: Extend JK and construct a perpendicular line a point J. hen construct a segment congruent HI on the perpendicular line.) age 10 of 30
11 Example Lesson 293: Explain the steps you would use to construct a tangent to circle C at a point. First use the straightedge to draw C. hen construct the perpendicular line to C that passes through point. he perpendicular line will be tangent to circle C. age 11 of 30
12 Name class date Geometry Unit 4 ractice Lesson In the diagram shown, is tangent to circle, the radius of the circle is 7 units, and 5 24 units. Find. 1. Use the diagram shown. C H D E F G 7 a. Identify a line that is tangent to the circle. 4. ttend to precision. Line WX is tangent to circle at point X. Line W intersects the circle at points and Q. he radius of circle is 9 units and W 5 6 units. What is WX? b. Identify a radius of the circle. c. Identify a chord of the circle. Q d. Which segment is perpendicular to the tangent line? 2. oint is a point on circle O. Which statement is NO true? X. O is a radius of the circle. W. here are many chords of the circle that contain point. 5. eason quantitatively. In this diagram, the radius of circle M is 10 and S 5 SQ 5 8. What is the length SN? C. here are many tangent lines that contain point. Q N D. here is exactly one diameter that contains point. 8 S 8 M College oard. ll rights reserved. age 12 of 30 1 Springoard Geometry, Unit 4 ractice
13 Name class date Lesson Make use of structure. he distance between the center of a circle and a chord is 15 cm. a. If the radius of the circle is 20 cm, what is the length of the chord? b. If the length of the chord is 20 cm, what is the radius of the circle? 10. eason quantitatively. In the diagram, chords Q and S intersect at point S 2 Q 7. he length of a chord of a circle is 21.2 cm and that chord is 5 cm from the center of the circle. a. What is the length of the diameter of the circle? b. What is the length of a chord of the circle that is 3 cm from the center of the circle? a. If 5 8, Q 5 3, and 5 6, what is S? b. If Q 5 12, Q 5 2, and 5 5, what is S? c. If 5 m, Q 5 n, and S 5 p, what is in terms of m, n, and p? d. If 5 S and 5 2? Q, what is Q? 8. hink about a chord and a diameter of the same circle. a. How are they similar? Lesson In the diagram shown, M and N are tangent to circle. M Q b. How can they be different? 9. Which of the following statements is NO true?. If a radius is perpendicular to a chord, it bisects the chord.. If two chords are perpendicular, one of them must bisect the other. C. If a diameter is perpendicular to a chord, it bisects the chord. D. ny two distinct diameters of a circle bisect each other. a. If M 5 16 and M 5 15, what is Q? b. If M 5 24 and Q 5 8, what is N? N c. If Q 5 a and Q 5 b, write an expression for N in terms of a and b. 2 d. If M 5 12 and Q 5 8, what is M? 2015 College oard. ll rights reserved. Springoard Geometry, Unit 4 ractice age 13 of 30
14 Name class date 14. Construct viable arguments. his diagram can be used to prove the theorem about two tangents to a circle from a point outside the circle. 12. In the diagram shown 5 C, 5 15, C 5 4 and, C, and C are tangent to circle X. S N M Q X Q a. How are the sides of SQ related to each other? b. What kind of figure is SQ? Explain. C c. How are angles S and Q related to each other? Explain. a. Find the perimeter of C. d. How are angles SQ and SQ related to each other? Explain. b. If the radius of circle X is 2 units, what is X? Write your answer as a radical. e. How are segments SQ and related to each other? 15. Make sense of problems. In the diagram, the three segments are tangent to the circle. DE 5 17, DF 5 12, and DH = Which statement about tangents to a circle is NO true?. tangent to a circle is perpendicular to a radius at the point of tangency. D. If two segments are tangent to a circle from the same point outside the circle, the ray from to the center of the circle bisects the angle formed by the two tangents. J C. If two segments from the same point outside a circle are tangent to the circle, then the line joining the two points of tangency can be a diameter of the circle. D. If is tangent to circle O at point, and Q is any point on other than, then there is another line through Q that is tangent to circle O. K E H G F a. Find the perimeter of DEF. b. If the radius of the circle is 3, find the distance from the center of the circle to point E to the nearest tenth College oard. ll rights reserved. age 14 of 30 3 Springoard Geometry, Unit 4 ractice
15 Name class date 18. Which of the following statements is true? Lesson he two radii that form a major arc can also form a diameter. 16. Make use of structure. In circle, m D 5 60, C bisects D, and is a diameter.. minor arc, plus a major arc, can form a full circle. C. he total measure of a major arc and a minor arc can be 180. D. major arc and a minor arc can form a semicircle. C 19. In the diagram of circle with diameter, m C 5 (5x 2 7) and m C 5 12x. D C a. What is m C? 5x x b. What is m CD? c. Identify three major arcs. d. Name three adjacent arcs that form a semicircle. a. Find x. 17. In circle D, m D Find each measure. b. Find m C and m C. 20. Model with mathematics. In the diagram shown, and are tangent to circle C at points and. he measure of is 36, and is a point on. major arc 388 D Q S D C a. m b. m S a. Find m C. c. m SQ. b. Find m D d. m QD c. Find m D. e. m SQ 2015 College oard. ll rights reserved. age 15 of 30 d. Find m. 4 Springoard Geometry, Unit 4 ractice
16 Name class date 23. square is inscribed in a circle. If the area of the square is 49 square units, what is the radius of the circle? Lesson eason quantitatively. In circle, and CD are diameters, m CDE 5 28, and m D Find each measure.. 2 unit units 2 C E C. 7 2 units D units 24. In circle, Q and S are diameters, and m Q D a. m EC b. m C c. m CD d. m C 608 Q S e. m C 22. In the diagram shown, S is a diameter of circle, 5 85, and m 5 40, m Q Find m S each measure. a. What is the specific name for quadrilateral QS? Q b. What is the measure of Q? S c. What is the measure of SQ? d. Use arc lengths to explain why m S 5 m QS. a. m S b. m Q c. m S d. mqs e. mq 2015 College oard. ll rights reserved. age 16 of 30 5 Springoard Geometry, Unit 4 ractice
17 Name class 25. ersevere in solving problems. In circle S, chords C and D intersect at E. Chord forms a 21 angle with chord C and m E eason quantitatively. In the circle shown, 5 61, m QM 5 110, and m 5 mq. m E S date 618 M C 1108 D Q? a. What is m C a. Find m MQ. b. What is m SC?. b. Find m Q c. What is m E?? d. What is m D c. What is m? e. ED EC because they are vertical angles. What is the measure of each angle? d. What is m? 28. In circle, C is a diameter, E, ED, and DC are 5 65, and m QC chords, m E Lesson (7x 2 10), m ZX In the circle shown, m XW 5 7x, and myw (11x 1 10), m ZY (11x 110)8 C X (7x 2 10)8 Z W x8 Q 658 D E 1358 Y a. What is m EC? a. Find x. b. Find m ED., m ZX, and m ZY. b. Find m XW c. What is m XW? c. Find m DC. d. If Z 5 y 1 7, W 5 3, Y 5 5, and X 5 2y, find Z. d. In QED, is QE QD? Explain College oard. ll rights reserved. age 17 of 30 6 Springoard Geometry, Unit 4 ractice
18 Name class 29. Suppose chords and CD intersect at point E inside the circle. Which of the following CNNO be true? can be congruent. and C. D date Lesson Express regularity in repeated reasoning. In the diagram shown, and are tangent to circle C. can be supplementary. and D. C can be complementary. and C C. C can form a semicircle. and C D. D D C 30. Construct viable arguments. In circle, diameter S is parallel to chord MN. Chords N and MS intersect at point. ell whether each statement is always, sometimes, or never true. 5 80, find m D and m. a. If m and m , find m b. If m D and m. c. If m 5 50, find m D S, find m D, and m., m d. If m 5 m M N 5 (3y 2 1), 5 (8y 1 5), m C 32. In circle O, m 5 (4y 1 13), and m D 5 (5y 1 3). m CD a. Figure SNM is a parallelogram. (8y15)8 b. Figure SNM is an isosceles trapezoid. (3y21)8 O c. S is obtuse. C (5y 13)8 d. SM and SNM are supplementary. D (4y 113)8 a. What is the value of y? b. Find m C. c. Find m D. d. What is the measure of? 2015 College oard. ll rights reserved. age 18 of 30 7 Springoard Geometry, Unit 4 ractice
19 Name class date 33. ersevere in solving problems. In the diagram shown, S is tangent to circle and intersects circle Q at and S. X is tangent to circle at and intersects circle Q at V and X. he two circles are tangent at point Y. lso, m SX and m S 34. In the diagram, is tangent to circle and D is a secant. 468 Q Y Z Q 1258 C D V X a. If m D and m C 5 80, find m CD and m. a. What is m V? b. If m 5 51 and m C 5 92, find m D and m CD. b. Find m SZX. c. If m CD and m C 5 81, find m D and m. c. What is m QY? d. If m 5 42 and m CD 5 160, find m C and m D. d. What is m Q? e. What is m Q? 35. he phrase "the measure of the angle is half the difference of the intercepted arcs" applies to all EXCE. an angle formed by two tangents.. an angle formed by two chords. C. an angle formed by two secants. D. an angle formed by a secant and a tangent College oard. ll rights reserved. Springoard Geometry, Unit 4 ractice age 19 of 30
20 Name class date Lesson Which statement describes the two steps necessary to prove that point Q is the midpoint of?. Show that Q 5 Q; show that Q 5 2? and Q 5 2?.. Show that Q 5 Q; show that Q and Q C. Show that, Q, and are collinear; show that Q 1 Q 5. D. Show that Q 5 Q; show that Q Q. 37. Suppose that points M and N are on a horizontal line, the coordinates of point M are (x 1, y 1 ), and the midpoint of MN is. a. Which ordered pair can you use for point N, (x 1, y 2 ) or (x 2, y 1 )? Explain. 39. eason abstractly. he center of a circle is C(a, b) and one endpoint of a diameter is D(2a, 2b). a. Find the coordinates of the other endpoint of that diameter. b. he diameter is divided into four congruent segments. Find the coordinates of the 5 points that determine those four congruent segments. 40. rectangle has one vertex at (0, 0) and its diagonals intersect at the point (m, n). a. What are the coordinates for the other three vertices of the rectangle? b. What is the distance from any vertex to (m, n)? c. What is the perimeter of the rectangle? b. What are the coordinates of point? d. What is the area of the rectangle? c. Use the Distance Formula to represent the lengths M and N. d. Use the Distance Formula to represent MN. e. Does M 5 N? Explain. 38. Express regularity in repeated reasoning. Find the coordinates of the midpoint of S for each pair of coordinates. a. (2a, 2b), S(2c, 2d) b. (24a, 26b), S(4a, 6b) c. (p, q), S(t, r) d. (a 1 3b, 3a 2 b), S(3a 1 5b, 5a 1 7b) Lesson Which statement about the slope of a line is NO true?. line that goes up from left to right has a positive slope.. line that goes down from left to right has a negative slope. C. horizontal line has a slope of zero. D. vertical line has a slope of Which of the following statements is true?. wo horizontal lines cannot be parallel to each other.. Every horizontal line is perpendicular to any vertical line. C. wo vertical lines can be perpendicular to each other. D. line with a positive slope can be parallel to a line with a negative slope College oard. ll rights reserved. Springoard Geometry, Unit 4 ractice age 20 of 30
21 Name class date 43. In the diagram, C DE and D is a vertical line. Complete the steps to show that parallel lines have equal slopes. y 44. eason abstractly. In the diagram, line m goes through (0, 0) and (a, b), and line n goes through (2b, a) and (0, 0). y m 1 n (2b, a) 1 4 (a, b) b D F 2 E C x D a b 2 C(0, 0) 3 a E x a. Why is 1 2? a. What is the relationship between DC and EC? Explain. b. What is the relationship between 2 and 3? Explain. c. What is m C? Explain. b. Why is FC DFE? d. Find the slopes of lines m and n. Show your work. e. Find the product of the slopes of lines m and n. Show your work. c. What does F FC represent? 45. Express regularity in repeated reasoning. wo lines p and q are perpendicular. Describe line q for each description of line p. a. Line p is horizontal. d. What does DF FE represent? b. Line p has a negative slope. c. Line p goes up from left to right. e. Why do C and DE have equal slopes? d. Line p has an undefined slope. e. Line p goes down from left to right College oard. ll rights reserved. Springoard Geometry, Unit 4 ractice age 21 of 30
22 Name class date Lesson 263 Use the diagram for Items hese items take you through a confirmation that the three medians of a specific triangle are concurrent. y ttend to precision. he third step is to find the point where two medians intersect. a. Using the equations for X and Y, solve each equation for y and set them equal to each other. Y 5 C5(0, 6) X b. Using your equation from art a, solve for x. (Hint: You can multiply both sides by a value to remove the fractions.) Show your work. 25 5(24, 0) Z 5 5(8, 0) 10 x he first step is to find the midpoints of the sides. a. What are the coordinates of X, the midpoint of C? Show your work. c. Using the value for x from art b and the equation for X or Y, find the corresponding yvalue for the point of intersection of the medians. Show your work. b. What are the coordinates of Y, the midpoint of C? Show your work. d. Write the coordinates of the point of intersection of X and Y. c. What are the coordinates of Z, the midpoint of? Show your work. d. What is the effect of having even integers for the coordinates of points,, and C? 47. he second step is to write an equation for each median. a. Find the slopes of X, Y, and CZ. Show your work. b. Use the slopes and points,, and C to write equations for each median in pointslope form. 49. Make use of structure. he last step is to show that the intersection of two medians is a point that is on the third median. a. Show that the point of intersection of X and Y is on CZ. Show your work. b. Summarize what you did in Items College oard. ll rights reserved. Springoard Geometry, Unit 4 ractice age 22 of 30
23 Name class date 50. student is proving that the medians of DEF are concurrent. So far, the student has found equations for the three medians DK, EJ, and FH. Which can be the next steps in the student's proof? y 52. In each set of three ordered pairs, and are the endpoints of a segment and is a point on that segment. Show that and have the same slope, and then find the ratio :. a. (2, 8), (8, 10), (5, 9) D b. (23, 9), (5, 27), (3, 23) J H 53. oints X,, and Y are on a line segment. Which of the following statements is NO correct? F K E x y Y. Find the slopes of two medians, and show that the product of the slopes is 21.. Decide whether the three medians intersect inside, on, or outside the triangle, and illustrate each of those with a separate diagram. C. Find the point of intersection for FH and EJ, and then show that DK contains that point. D. Find the point of intersection for FH and EJ, and then find the distance from that point to the three vertices of the triangle. Lesson Model with mathematics. For each pair of ordered pairs, X and Y, find the coordinates of a point that lies 3 of the way from X to Y. 4 a. X(0, 0), Y(20, 28) X 3. is 60% of the distance from X to Y.. he ratio X : Y is 3 : 2. C. he ratio Y : XY is 3 : 5. D. X and Y have the same slope. 54. oint H lies along a directed line segment from J(5, 8) to K(1, 1). oint H partitions the segment into the ratio 7 : 3. Find the coordinates of point H. 2 x b. X(5, 1), Y(13, 25) c. X(10, 1), Y(2, 23) 55. ttend to precision. Find the coordinates of point M that divides the directed line segment from (21, 3) to Q(9, 8) and partitions the segment into the ratio 4 to 1. d. X(5, 23), Y(11, 218) College oard. ll rights reserved. Springoard Geometry, Unit 4 ractice age 23 of 30
24 Name class date Lesson circle has the equation (x 1 5) 2 1 (y 2 8) a. Find two points on the circle that have xcoordinate circle contains points (2, 1), (6, 1), and C(6, 5). 5 y C(6, 5) b. Find two points on the circle that have ycoordinate (2, 1) (6, 1) 5 10 x 57. Express regularity in repeated reasoning. Write an equation for each circle. a. center (5, 0) and radius a. Write the equation for the perpendicular bisector of. b. center (2, 23) and diameter 12 b. Write the equation for the perpendicular bisector of C. c. center (0, 24) and contains the point (6, 24) c. Using to label the intersection of the lines in arts a and b, what are the coordinates of? d. diameter has endpoints (25, 2) and (10, 10) 58. Make use of structure. Identify the center and radius for each circle. a. (x 2 3) 2 1 (y 2 2) b. (x 1 7) 2 1 (y 1 11) c. (x 2 3.8) 2 1 (y 2 1.2) d. (x 2 a) 2 1 (y 1 b) 2 5 m d. Find the distance from point to each of points,, and C. e. Write an equation for the circle with center and radius. 60. circle is drawn on a coordinate grid. Which statement is true?. Every line with a positive slope must either intersect the circle or be tangent to it.. ny line must meet the circle at either 0, 1, or 2 points. C. he distance between two points on the circle can never be greater than the length of the circle's radius. D. If two points are inside the circle, the distance between them can be greater than the length of the circle's diameter College oard. ll rights reserved. Springoard Geometry, Unit 4 ractice age 24 of 30
25 Name class date Lesson dd a term so that each expression is the square of a binomial. hen write the new expression in the form (x 1 a) 2. a. x 2 1 6x b. x x c. y 2 1 5y d. y y e. x 2 1 2a 62. Model with mathematics. Write each equation in the form (x 1 a) 2 1 (y 1 b) 2 5 c. hen tell what number you added to each side of the original equation. a. x 2 1 6x 1 y 2 1 4y 5 0 b. x 2 2 4x 1 y y 5 0 c. x 2 1 2x 1 y 2 2 5y 5 1 d. x 2 2 9x 1 y Construct viable arguments. Determine if the equation represents a circle. If it does, tell the center of the circle. a. x 2 1 6x 1 y 2 2 7y 5 1 b. x 2 2 7x 1 5y 5 35 c. x 2 1 y 2 1 3y 5 5 d. (x 2 5) 2 1 (2y 1 3) he equation x 2 1 3x 5 y 2 2 4y 5 6 represents a circle. In which quadrant is the center of the circle?. Quadrant I. Quadrant II C. Quadrant III D. Quadrant IV Lesson In the diagram, is the point (0, t) and m is a horizontal line. Which expression represents the distance between point and any point (x, y) on line m? (0, t) y Find the center and radius of the circle represented by each equation. 25 m (x, y) 5 x a. (x 1 5) 2 1 y y b. x 2 1 8x 1 y 2 2 2y 5 8 c. x x 1 y 2 2 6y 5 29 d. x 2 1 x 1 y 2 2 y x 2 1 (y 2 t) 2. x 1( y2t) 2 2 C. (x 2 t) 2 1 y 2 D. x5( y2t) 2 1y College oard. ll rights reserved. Springoard Geometry, Unit 4 ractice age 25 of 30
26 Name class date 67. parabola opens up or down. Write the equation of the parabola for the given information. a. focus (0, 5), directrix y 5 5 b. focus (0, 3), directrix y 5 23 c. vertex (0, 0), focus (0, 22) d. directrix y 5 26, vertex (0, 6) 68. Model with mathematics. Write the focus and directrix for each parabola. a. opens to the left; focus (27, 0); directrix x 5 7 b. opens down; focus (0, 21); directrix y Make use of structure. For each parabola described below, the vertex of the parabola is the origin. Write the coordinates of the focus. 1 2 a. y 5 x b. y 5 x c. x 5 2 y 10 1 d. x 5 y 2 r Lesson What is the vertex of each parabola? a. y (x 2 3)2 8 c. opens up; focus 0, 1 4 ; directrix y d. opens to the right; focus (2.5, 0); directrix x b. y 5 1 (x 1 5)2 10 c. x (y 1 2) Write an equation for the parabola illustrated in the diagram. d. y 5 t 1 (x 1 a) 2 1 b 5 y directrix: y m focus: (0, 21.25) 5 x College oard. ll rights reserved. Springoard Geometry, Unit 4 ractice age 26 of 30
27 Name class date 72. Construct viable arguments. Complete the steps to write the equation for the parabola with vertex (21, 5) and directrix y y 5 focus: (21, 5) 25 directrix: y521 5 x 74. Which statement describes the vertex and directrix of the parabola y (x 1 5)2?. vertex (25, 3); directrix y vertex (25, 3); directrix y 5 1 C. vertex (3, 25); directrix y 5 29 D. vertex (3, 25); directrix y he equation of a parabola is y 2 1 2y x 5 5. Find the vertex of the parabola. Explain your steps. a. Describe how to find the vertex for the parabola. hen find the vertex. b. Explain how to find p for the parabola. hen find p. Lesson Use appropriate tools strategically. Use quadrilateral QS. Do not erase your construction marks. Q Q c. ell which way the parabola opens, and explain how you know. S S O O d. Write the general form for the equation of the parabola. hen give the values of h, k, and p. a. Using O, construct O so O 5 Q. e. Use the information in art d to write the equation for the parabola. 73. Make use of structure. For each parabola, (i) Determine the direction of the opening. (ii) Determine whether the general form for the 1 parabola is y 2 k 5 (x 2 h) 4p 2 or 1 y 2 h 5 (y 2 k) 2. 4p (iii) Give the values of h, k, and p for the parabola. (iv) Write the equation for the parabola. b. Draw CD. hen identify XY on CD, so XY 5 S. c. Construct a segment whose length is S 2 S. Label that segment "S 2 S." d. Construct a segment whose length is three times Q. a. vertex (23, 1), directrix x 5 26 b. focus (0, 1.5), directrix y College oard. ll rights reserved. Springoard Geometry, Unit 4 ractice age 27 of 30
28 Name class 77. Use C. Do not erase your construction marks. 80. One side of XYZ has a length of 17 cm. Which pairs of lengths CNNO be the lengths of the other two sides of the triangle? C date. 1 cm, 17 cm C. 35 cm, 19 cm X X Y C. 25 cm, 5 cm Y D. 10 cm, 10 cm a. Using XY, copy angle at point X. Lesson Use appropriate tools strategically. hrough point H, construct a line p that is parallel to line m. hen, through point Q, construct a line q that is also parallel to line m. b. oint is on DE. Copy angles,, and C so they are nonoverlapping adjacent angles and each has vertex. D E H m 78. Construct the perpendicular bisector of each given segment. Do not erase your construction marks. a. b. M Q N 82. Construct a line r that contains point C and is perpendicular to. hen select a point D on line r and, through D, construct a line s that is perpendicular to line r. C S 79. ttend to precision. Construct the angle bisectors for each angle of S. S 2015 College oard. ll rights reserved. age 28 of Springoard Geometry, Unit 4 ractice
29 Name class HK and point N is 83. In the diagram, point M is on not on HK. H date 85. What construction is shown in the diagram? N M K a. Construct line a through point N so that a HK. b. Construct line b through point M so that b HK.. finding the perpendicular bisectors of the three sides of a triangle c. What is the relationship between lines a and b?. finding the bisectors of the three angles of a triangle d. Construct line c through point N so that c a. C. finding the medians to the three sides of a triangle D. finding the altitudes to the three sides of a triangle e. What is the relationship between line c and HK? Lesson eason abstractly. oints, S, and are on a circle. 86. Use appropriate tools strategically. is the radius of a circle. Construct a circle with that radius. hen construct a regular inscribed hexagon in the circle. S a. Draw S. hen construct line m so it is the perpendicular bisector of S. 87. is a diagonal of square QS. Construct that square. b. Draw S. hen construct line n so it is the perpendicular bisector of S. c. How is the intersection of m and n related to the circle? d. Use a compass to verify your answer to art c College oard. ll rights reserved. age 29 of Springoard Geometry, Unit 4 ractice
30 Name class date 90. student has constructed square V inscribed in circle N. he student wants to inscribe a regular octagon in the circle. Which construction will NO result in the other four vertices of the octagon? 88. ttend to precision. Follow these steps to inscribe a circle in MN. M N N a. isect N. Use s to label the bisector. V b. Find the intersection of the bisectors of N and. Use to label that point.. Using as the center and N as a radius, draw arcs on the circle on each side of. epeat using as the center. c. Construct line z through point so that z is perpendicular to N. Use Q to label the intersection of line z and N.. Construct the perpendicular bisectors of V and, and identify the four points where the perpendicular bisectors intersect the circle. C. Draw diameters and V for the circle. isect the four central angles, and identify the points where the angle bisectors intersect the circle. d. Using as the center and Q as a radius, construct circle. 89. oint is a point on circle. D. Construct perpendiculars from point N to each of the four sides of the square. Identify the points where the perpendiculars intersect the circle. a. Construct line t that is tangent to circle at point. Explain your steps. b. Construct circle that is tangent to circle. Circle should have a center that is on and a radius equal to the radius of circle. Explain your steps College oard. ll rights reserved. age 30 of Springoard Geometry, Unit 4 ractice
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