Study Guide and Review

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1 Fill in the blank in each sentence with the vocabulary term that best completes the sentence. 1. A is a flat surface made up of points that extends infinitely in all directions. A plane is a flat surface made up of points that extends infinitely in all directions. Therefore, the correct answer is plane. 2. A set of points that all lie on the same line are said to be. A set of points that all lie on the same line are said to be collinear points. Therefore, the correct answer is collinear. 3. If two lines intersect to form four right angles, the lines are called. If two lines intersect to form four right angles, the lines are called perpendicular. 4. If the sum of the measures of two angles is 180, then the angles are called angles. If the sum of the measures of two angles is 180, then the angles are called supplementary angles. Use the figure to complete each of the following. 6. Give another name for line b. There are two point S and T marked on the line b. So, the line b can also be called 7. Name a point that is not contained in any of the three lines a, b, or c. Here, W is a point on the plane R and is not on any of the lines on the plane. 8. Give another name for plane WPX. 9. The plane WPX is also named as plane R. Name the geometric term that is best modeled by each item. In the figure, two flat surfaces intersect each other. The two flat surfaces model two planes. Since the intersection of two planes form a line, the item models a line. 5. Name the intersection of lines a and c. The lines a and c intersect at the point P. 10. Each bead denotes a location. So, it models a point. esolutions Manual - Powered by Cognero Page 1

2 Find the value of the variable and XP, if X is between P and Q. 11. XQ = 13, XP = 5x 3, PQ = 40 Here X is between P and Q. So, PQ = XP + XQ. 13. Determine whether each pair of segments is congruent. We have XQ = 13, XP = 5x 3, and PQ = = x 3 30 = 5x 6 = x So, XP = 5(6) 3 = Segments that have the same measure are called congruent segments. Here, the lengths of the segments AB, BC, CD, and AD are given to be equal. So, they are congruent. 12. XQ = 3k, XP = 7k 2, PQ = 6k + 16 Here X is between P and Q. So, PQ = XP + XQ. We have XQ = 3k, XP = 7k 2, PQ = 6k k + 16 = 7k 2 + 3k 6k 10k = k = 18 k = 4.5 Segments that have the same measure are called congruent segments. Substitute for a to find the lengths. YZ = 3a + 2 = 3(4) + 2 = 14 YX = 4a = 4(4) = 16 The lengths of the segments XY and YZ are not equal. So, they are not congruent. So, XP = 7(4.5) 2 = esolutions Manual - Powered by Cognero Page 2

3 15. DISTANCE The distance from Salvador s job to his house is 3 times greater than the distance from his house to school. If his house is between his job and school and the distance from his job to school is 6 miles, how far is it from Salvador s house to school? Salvador s house is between his job and school. Let x be the distance from his house to school. Then the distance from his job to his house is 3x. The distance from his job to school is 6 miles. 17. P(2, 1), Q(10, 7) Use the Distance Formula. Substitute. That is, 6 = 3x + x. 6 = 4x 1.5 = x Therefore, the house is 1.5 miles away from the school. Find the distance between each pair of points. 16. A( 3, 1), B(7, 13) Use the Distance Formula. Substitute. The distance between P and Q is 10 units. Find the coordinates of the midpoint of a segment with the given endpoints. 18. L( 3, 16), M(17, 4) Use the Midpoint Formula. Substitute. The midpoint of is (7, 10). The distance between A and B is or about 15.6 units. esolutions Manual - Powered by Cognero Page 3

4 19. C(32, 1), D(0, 12) Use the Midpoint Formula Substitute.. Find the coordinates of the missing endpoint if M is the midpoint of 20. X( 11, 6), M(15, 4) Let the coordinates of Y be (x, y). Then by the Midpoint Formula,. The midpoint of is (16, 6.5). Write two equations to find the coordinates of Y. The coordinates of Y are (41, 14). esolutions Manual - Powered by Cognero Page 4

5 21. M( 4, 8), Y(19, 0) Let the coordinates of X be (x, y). Then by the Midpoint Formula, Write two equations to find the coordinates of X HIKING Carol and Marita are hiking in a state park and decide to take separate trails. The map of the park is set up on a coordinate grid. Carol s location is at the point (7, 13) and Marita is at (3, 5). a. Find the distance between them. b. Find the coordinates of the point midway between their locations. a. Use the Distance Formula. Substitute. The coordinates of X are ( 27, 16). The distance between Carol and Marita is b. Use the Midpoint Formula. or about 8.9 units Substitute. The midpoint of the locations of Carol and Marita is (5, 9). esolutions Manual - Powered by Cognero Page 5

6 For Exercises 23 26, refer to the figure below. 27. SIGNS A sign at West High School has the shape shown. Measure each of the angles and classify them as right, acute, or obtuse. 23. Name the vertex of 7. Here, 7 is same as the angle CGJ. So, the vertex of the angle is G. 24. Write another name for 4. The angles A and B measure 90. So, they are right angles. The angles E and C measure more than 90. So, they are obtuse angles. The angle D measures less than 90. So, it is an acute angle. For Exercises 28 30, refer to the figure below. Here, 4 is same as the CDG. 25. Name the sides of 2. Here, 2 is same as the ACH. Therefore, its sides are 26. Name a pair of opposite rays. Any point on a line determines exactly two opposite rays. Here, are two opposite rays. 28. Name an angle supplementary to TVY. Supplementary angles are two angles with measures that have a sum of 180. Here, TVR is supplementary to TVY. 29. Name a pair of vertical angles with vertex W. Vertical angles are two nonadjacent angles formed by two intersecting lines. Here, QWP and XWV are a pair of vertically opposite angles. esolutions Manual - Powered by Cognero Page 6

7 30. If m SXW = 5x 16, find the value of x so that Since That is, 5x 16 = 90. Add 16 to both sides. 5x = 106 Divide both sides by 5. x = PARKING The parking arm shown below rests in a horizontal position and opens to a vertical position. After the arm has moved 24, how many more degrees does it have to move so that it is vertical? 32. Name each polygon by its number of sides. Then classify it as convex or concave and regular or irregular. The polygon has 3 sides. So, it is a triangle. No line containing any of the sides will pass through the interior of the triangle, so it is convex. All of the sides are congruent, so it is equilateral. All of the angles are congruent, so it is equiangular. Since the polygon is convex, equilateral, and equiangular, it is regular. So this is a regular triangle. 33. The angle between the initial position of the arm and the vertical position will be 90. So, once it is moved 24, to reach 90 it has to be moved (90 24) or 66 more. The polygon has 12 sides. So, it is a dodecahedron. Four of the lines containing the sides of the polygon will pass through the interior of the dodecahedron, so it is concave. Only convex polygons can be regular, so this is an irregular dodecahedron. esolutions Manual - Powered by Cognero Page 7

8 34. Find the perimeter of quadrilateral ABCD with vertices A( 3, 5), B(0, 5), C(2, 0), and D( 5, 0). Use the Distance Formula of each side of the quadrilateral. to find the length Therefore, the perimeter of the quadrilateral is or about 20.8 units. 35. PARKS Westside Park received 440 feet of chain-link fencing as a donation to build an enclosed play area for dogs. The park administrators need to decide what shape the area should have. They have three options: (1) a rectangle with length of 100 feet and width of 120 feet, (2) a square with sides of length 110 feet, or (3) a circle with radius of approximately 70 feet. Find the areas of all three enclosures and determine which would provide the largest area for the dogs. The area of a rectangle of length l and width w is l w. So, the area of a rectangle of length 100 feet and width 120 feet is 100(120) = 12,000 ft 2. The area of a square of side length s is s 2. So, the area of a square of side length 110 feet is (110) 2 = 12,100 ft 2. The area of a circle of radius r is πr 2. So, the area of a square of radius 70 feet is Therefore, the third option provides the largest area. esolutions Manual - Powered by Cognero Page 8

9 Identify each solid. Name the bases, faces, edges, and vertices. Find the surface area and volume of each solid. 36. The base of the pyramid is a square. So, it is a square pyramid. The other faces are triangles. The base is The other faces are The edges are The vertices are A, B, C, D, and X. 38. The surface area of a cylinder of radius r and height h is given by the formula, Here, the radius of the cylinder is 6 cm and the height is 10 cm. So, the area is Therefore, the area of the cylinder is about cm The bases of the prism are hexagons. So, it is a hexagonal prism. The other faces are rectangles. The bases are ABCDEF and GHJKLM. The other faces are The volume of a cylinder of radius r and height h is given by the formula, Here, the radius of the cylinder is 6 cm and the height is 10 cm. So, the volume is The edges are The vertices are A, B, C, D, E, F, G, H, J, K, L, and M. Therefore, the volume of the cylinder is about cm 3. esolutions Manual - Powered by Cognero Page 9

10 Therefore, the volume of the pyramid is 384 in The surface area of a pyramid of regular base with an area B, base perimeter P, and slant height is given by the formula, Here, the height of the pyramid is 8 in. and each side of the base is 12 in. So, the slant height can be found using the Pythagorean Theorem. 40. The surface area of a cylinder of radius r, slant height, and height h is given by the formula, Here, the radius of the cylinder is 3 ft, the slant height is 5 ft, and the height is 4 ft. So, the area is So, the slant height of the pyramid is 10 in. Then, the area is Therefore, the area of the cone is about 75.4 in 2. The volume of a cone of radius r and height h is given by the formula, Therefore, the area of the pyramid is 384 in 2. The volume of a pyramid of base area B and height h is given by the formula, Here, the radius of the cylinder is 3 ft. and the height is 4 ft. So, the volume is Here, the base area of the pyramid is 144 in 2 and the height is 8 in. So, the volume is Therefore, the volume of the cone is about 37.7 ft 3. esolutions Manual - Powered by Cognero Page 10

11 42. BUILDING Chris is building a trunk like the one shown below. His design is a square prism. What is the volume of the trunk? 41. The triangular prism has isosceles triangles as bases and rectangular lateral faces. The total surface area of the prism is the sum of the areas of the bases and the lateral faces. The altitude of each base is 4 m and the length of the base is 6 m. So, the area of each base is Each of the lateral faces has a width of 3 m and the lengths 6 m, 5 m, and 5 m respectively. So, the area of the lateral surfaces is The volume of a prism of base area B and height h is given by the formula, Here, the area of each base of the prism is 3(3) = 9 ft 2 and the height is 2 ft. So, the volume is Therefore, the volume of the trunk is 18 ft 3. Therefore, the total surface area is The volume of a prism of base area B and height h is given by the formula, Here, the area of each base of the prism is 12 m 2 and the height is 3 m. So, the volume is Therefore, the volume of the prism is 36 m 3. esolutions Manual - Powered by Cognero Page 11

12 43. HOCKEY A regulation hockey puck is a cylinder made of vulcanized rubber 1 inch thick and 3 inches in diameter. Find the surface area and volume of a hockey puck. The surface area of a cylinder of radius r and height h is given by the formula, Here, the radius of the hockey puck is 1.5 in. and the height is 1 in. So, the area is Therefore, the surface area of the hockey puck is about 23.6 in 2. The volume of a cylinder of radius r and height h is given by the formula, Here, the volume is Therefore, the volume of the hockey puck is about 7.1 in 3. esolutions Manual - Powered by Cognero Page 12