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Transcript
ACT Geometry Practice Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. Refer to Figure 1. Figure 1 ____ ____ 1. Which is these is NOT a way to refer to line BD? a. c. b. m d. line JD 2. Are points A, C, D and F coplanar? Explain. a. b. c. d. Yes; they all lie on plane P. No; they are not on the same line. Yes; they all lie on the same face of the pyramid. No; three lie on the same face of the pyramid and the fourth does not. Refer to Figure 2. Figure 2 ____ ____ 3. Name a point that is NOT coplanar with G, A, and B. a. K c. C b. D d. F 4. Name an intersection of plane GFL and the plane that contains points A and C. a. line LC c. line AC b. C d. plane CAB In the figure, ____ 5. If a. 33 b. 58 bisects and . , find x. c. 11 d. 29 In the figure, ____ ____ ____ and are opposite rays. 6. Which is NOT true about ? a. is acute. b. c. Point M lies in the interior of d. It is an angle bisector. 7. If and a. 137 b. 12 8. . Point R lies on and . . , what is ? c. 4.2 d. 43 . If a. 11.5 b. 7.83 bisects , find f. c. 3.5 d. 19 Name each polygon by its number of sides. Then classify it as convex or concave and regular or irregular. ____ 9. a. triangle, convex, regular b. triangle, concave, irregular c. triangle, convex, irregular d. quadrilateral, convex, irregular Find the length of each side of the polygon for the given perimeter. ____ 10. in. Find the length of each side. ____ a. 11 in., 20 in., 35 in. b. 10 in., 18.5 in., 31.5 in. 11. cm. Find the length of each side. a. 8 cm, 8 cm, 8 cm b. 6 cm, 6 cm, 6 cm c. 12 in., 21.5 in., 38.5 in. d. 10 in., 15 in., 35 in. c. 10 cm, 10 cm, 4 cm d. 9 cm, 9 cm, 6 cm Determine whether the conjecture is true or false. Give a counterexample for any false conjecture. ____ 12. Given: point B is in the interior of . Conjecture: a. False; may be obtuse. b. True c. False; just because it is in the interior does not mean it is on the bisecting line. d. False; . ____ 13. Given: Conjecture: a. False; . c. False; b. True d. False; 14. Given: Two angles are supplementary. Conjecture: They are both acute angles. a. False; either both are right or they are adjacent. b. True c. False; either both are right or one is obtuse. d. False; they must be vertical angles. ____ . . ____ 15. Given: segments RT and ST; twice the measure of Conjecture: S is the midpoint of segment RT. a. True b. False; point S may not be on . c. False; lines do not have midpoints. d. False; could be the segment bisector of . Refer to the figure below. ____ ____ ____ 16. Name all planes intersecting plane CDI. a. ABC, CBG, ADI, FGH b. CBA, DAF, HGF c. BAD, GFI, CBG, GFA d. DAB, CBG, FAD 17. Name all segments parallel to a. b. c. d. 18. Name all segments skew to a. b. . . c. d. . ____ 19. In the figure, . Find x and y. a. b. c. d. Given the following information, determine which lines, if any, are parallel. State the postulate or theorem that justifies your answer. ____ 20. a. b. c. d. ; congruent corresponding angles ; congruent corresponding angles ; congruent alternate interior angles ; congruent alternate interior angles Find each measure. ____ 21. a. b. c. d. Name the congruent angles and sides for the pair of congruent triangles. ____ 22. a. b. c. d. Determine whether given the coordinates of the vertices. Explain. ____ 23. ____ a. Yes; two sides of triangle PQR and angle PQR are the same measure as the corresponding sides and angle of triangle STU. b. Yes; each side of triangle PQR is the same length as the corresponding side of triangle STU. c. No; one of the triangles is obtuse. d. No; each side of triangle PQR is not the same length as the corresponding side of triangle STU. 24. Triangles ABC and AFD are vertical congruent equilateral triangles. Find x and y. a. c. b. d. ____ 25. Triangles MNP and OMN are congruent equilateral triangles. Find x and y. a. b. ____ 26. c. d. is an altitude. a. 34 b. 32 , . Find . c. 18 d. 31 Determine the relationship between the lengths of the given sides. ____ 27. a. b. c. cannot be determined d. Determine whether the given measures can be the lengths of the sides of a triangle. Write yes or no. Explain. ____ 28. 3, 9, 10 a. Yes; the third side is the longest. b. No; the sum of the lengths of two sides is not greater than the third. c. No; the first side is not long enough. d. Yes; the sum of the lengths of any two sides is greater than the third. ____ 29. An isosceles triangle has a base 9.6 units long. If the congruent side lengths have measures to the first decimal place, what is the shortest possible length of the sides? a. 4.9 c. 4.7 b. 19.3 d. 9.7 Determine whether each pair of figures is similar. Justify your answer. ____ 30. a. b. is not similar to . Corresponding angles are not the same. because the corresponding angles of each triangle are congruent. The ratio of the corresponding sides is 1. c. is not similar to . The ratios of the corresponding sides are not the same. d. because the corresponding angles of each triangle are congruent. The ratio of the corresponding sides is 2. Find x and the measures of the indicated parts. ____ 31. AB and BC a. b. c. d. ____ 32. BD and CE a. c. b. d. Find the perimeter of the given triangle. Round your answer to the nearest tenth if necessary. ____ 33. , if , is a parallelogram, a. 32 b. 56 ____ ____ 34. Find AD if , , , and . c. 24 d. 13.7 , and are medians, , , , and . a. 15 c. 6 b. 48 d. 18 35. Dante is standing at horizontal ground level with the base of the Empire State Building in New York City. The angle formed by the ground and the line segment from his position to the top of the building is 48.4°. The height of the Empire State Building is 1472 feet. Find his distance from the Empire State Building to the nearest foot. a. 1307 c. 2217 b. 7.65 d. 1968 ____ 36. A space shuttle is one kilometer above sea level when it begins to climb at a constant angle of 3° for the next 80 ground kilometers. About how far above sea level is the space shuttle after its climb? a. 4.2 kilometers c. 79.9 kilometers b. 5.2 kilometers d. 80.9 kilometers A landscaper is making a retaining wall to shore up the side of a hill. To ensure against collapse, the wall should make an angle 75° or less with the ground. ____ 37. How far from the base of the hill is the base of a 15-foot slanted wall? a. 3.88 ft c. 55.98 ft b. 14.49 ft d. 57.96 ft A 60-yard long drawbridge has one end at ground level. The other end is initially at an incline of 5°. ____ ____ ____ 38. During one stage of the drawbridge’s motion, the raised end is 15 yards above the ground. What is the incline of the drawbridge to the nearest hundredth? a. 0.004° c. 14.48° b. 14.04° d. 75.52° 39. Two horses are observed by a hang glider 80 meters above a meadow. The angles of depression are 10.4° and 8°. How far apart are the horses? a. 133.3 m c. 569.2 m b. 435.9 m d. 1005.1 m 40. Zack, Rachel, and Maddie are unraveling a huge ball of yarn to see how long it is. As they move away from each other, they form a triangle. The distance from Zack to Rachel is 3 meters. The distance from Rachel to Maddie is 2.5 meters. The distance from Maddie to Zack is 4 meters. Find the measures of the three angles in the triangle. a. , , b. , , c. , , d. , , Complete the statement about parallelogram ABCD. ____ 41. a. b. c. d. ____ ; Alternate interior angles are congruent. ; Alternate interior angles are congruent. ; Opposite angles of parallelograms are congruent. ; Opposite angles of parallelograms are congruent. Determine whether a figure with the given vertices is a parallelogram. Use the method indicated. ____ 42. a. b. c. d. ____ , , , ; Distance and Slope Formulas no; The opposite sides are not congruent and do not have the same slope. yes; The opposite sides do not have the same slope. no; The opposite sides do not have the same slope. yes; The opposite sides are not congruent and do not have the same slope. 43. In rhombus YZAB, if a. 24 b. 12 12, find . c. 6 d. Given each set of vertices, determine whether parallelogram ABCD is a rhombus, a rectangle, or a square. List all that apply. ____ 44. , , , a. square; rectangle; rhombus b. rhombus c. square d. rectangle ____ 45. For trapezoid JKLM, A and B are midpoints of the legs. Find ML. ____ a. 4 c. 68 b. 34 d. 40 46. Find the exact circumference of the circle. a. 12 b. 24 mm mm c. 12 mm d. 6 mm Use the diagram to find the measure of the given angle. ____ 47. a. 50 b. 40 c. 60 d. 30 ____ 48. In , TS = 15, UQ = US. Find m a. 28 b. 30 ____ . c. 15 d. 39 49. If a. 40 b. 25 = 40, = 120, and = 110, find c. 50 d. 80 . ____ 50. Find x. Assume that segments that appear tangent are tangent. a. 7 b. 6 c. 14 d. 5 Find the measure of the numbered angle. ____ 51. a. 62.5 b. 105 c. 112.5 d. 115 Find x. Assume that any segment that appears to be tangent is tangent. ____ ____ 52. a. 12 b. 14 c. 8 d. 10 a. 35 b. 20 c. 25 d. 30 53. Find x. Round to the nearest tenth if necessary. ____ 54. a. 7 b. 8 c. 9 d. 10 Find x. Round to the nearest tenth if necessary. Assume that segments that appear to be tangent are tangent. ____ 55. a. 5 b. 2.8 c. 3.5 d. 4 ____ 56. ____ a. 3 c. 2 b. 7 d. 4 57. Find the perimeter and area of the parallelogram. Round to the nearest tenth if necessary. ____ a. 88 mm; 415.7 c. 44 mm; 415.7 b. 44 mm; 346.4 d. 88 mm; 346.4 58. Find the area of the figure. Round to the nearest tenth if necessary. a. 672 b. 74 c. 336 d. 1344 Find the area of the figure. Round to the nearest tenth if necessary. ____ 59. a. 148 units2 b. 140 units2 ____ ____ c. 130 units2 d. 122 units2 60. a. 541.9 units2 c. 192 units2 2 b. 624 units d. 315.8 units2 61. The solid below is a composite of a cube and a square pyramid. The base of the solid is the base of the cube. Find the lateral area of the solid. a. 80 b. 660 c. 560 d. 720 Find the surface area of the regular pyramid. Round to the nearest tenth if necessary. ____ 62. a. 496.7 b. 5184.0 ____ c. 864.0 d. 356.4 63. Find the lateral area of the cone. Use 3.14 for . Round to the nearest tenth if necessary. a. 35.0 b. 109.9 c. 188.4 d. 183.2 Determine whether each statement is true or false. If false, give a counterexample. ____ 64. If a plane intersects a sphere so that it contains the center of the sphere, then that intersection will sometimes be a great circle. a. True b. False, the intersection will never be a great circle. c. False, the intersection will always be a great circle. d. False, the intersection of a plane and a sphere cannot contain the center of the sphere. ____ ____ ____ ____ ____ 65. All radii of the same sphere are congruent. a. True b. False, no two radii of the sphere are congruent. c. False, some radii will be congruent, but not all. d. False, a segment joining two points on the sphere is a tangent. 66. Find the surface area of the sphere. Use 3.14 for . Round to the nearest tenth. a. 113 c. 28.3 b. 339.1 d. 36 67. Find the surface area of a sphere if the circumference of a great circle is 43.96 centimeters. Use 3.14 for . Round to the nearest tenth. a. 4308.1 c. 153.9 b. 196 d. 615.4 68. Suppose a snow cone has a paper cone that is 8 centimeters deep and has a diameter of 5 centimeters. The flavored ice comes in a spherical scoop with a diameter of 5 centimeters and rests on top of the cone. If all the ice melts into the cone, will the cone overflow? Explain. a. No. The volume of the ice is less than the volume of the cone. b. No. The volume of the ice is exactly the same as the volume of the cone. c. Yes. The volume of the ice is greater than the volume of the cone. d. There is not enough information given to solve this problem. 69. What is the volume of this sphere, rounded to the nearest tenth? a. 7234.6 cm3 c. 602.9 cm3 3 b. 1808.6 cm d. 150.7 cm3 Determine whether each statement is sometimes, always, or never true. ____ 70. Congruent spheres have equal surface areas. a. sometimes c. never b. always ACT Geometry Practice Answer Section MULTIPLE CHOICE 1. ANS: C The proper way to refer to a line is any 2 points on the line with an arrow above them or “line such-and-such”, where “such-and-such” is any 2 points on the line. Using three letters is not correct. TOP: Identify and model points¸ lines¸ and planes. KEY: Points, Lines, Planes NOT: /A/ Does line BD contain point J? /B/ Does that line contain points B and D? /C/ Correct! /D/ Are points J and D on line BD? 2. ANS: D Points that lie on the same plane are said to be coplanar. Three points are always coplanar but if the fourth point is not on the same plane with the first three, they are not all coplanar. TOP: Identify coplanar points and intersecting lines in space. KEY: Coplanar Points, Intersecting Lines, Lines in Space NOT: /A/ Do all four points lie on the same plane? Which plane? /B/ Do all four points lie on the same plane? Which plane?/C/ What does coplanar mean? /D/ Correct! 3. ANS: C Coplanar points are points that lie on the same plane. TOP: Identify planes in space. KEY: Planes, Planes in Space NOT: /A/ Is K in a different plane? /B/ What plane are you working with? /C/ Correct! /D/ What plane are you working with? 4. ANS: A The intersection of two planes is a line. TOP: Identify planes in space. KEY: Planes, Planes in Space NOT: /A/ Correct! /B/ Can the intersection of two planes be a point? /C/ Is point A on plane GFL? /D/ Can the intersection of two planes be a plane? 5. ANS: D Since bisects the equation to find x. , and . Solve for v, then substitute into either side of TOP: Identify and use congruent angles. KEY: Angles, Congruent Angles, Congruency NOT: /A/ Don’t forget to subtract./B/ You are not finding the measure of FGH. You are finding x. /C/ You are not finding v. You are finding x. /D/ Correct! 6. ANS: A so it is obtuse. TOP: Identify and use the bisector of an angle. KEY: Angle Bisectors NOT: /A/ Correct! /B/ If answer d is true, then this must be true. /C/ Being in the interior means being between the two end rays of an angle. /D/ If answer b is true, then this must be true. 7. ANS: D TOP: Identify and use the bisector of an angle. KEY: Angle Bisectors NOT: /A/ What angle are you looking for? /B/ Did you solve for q instead of the angle measure? /C/ What two angles added together equal angle LKN? /D/ Correct! 8. ANS: A Lines that form right angles are perpendicular. A right angle measures 90. TOP: Identify perpendicular lines. KEY: Perpendicular Lines NOT: /A/ Correct! /B/ Did you use the Distributive Property carefully? /C/ Check your math. /D/ What is the measure of the angle created by perpendicular lines? 9. ANS: C Suppose the line containing each side is drawn. If any of the lines contain any point in the interior of the polygon, then it is concave. Otherwise it is convex. A convex polygon in which all the sides are congruent and all the angles are congruent is called a regular polygon. TOP: Name polygons. KEY: Polygons, Name Polygons NOT: /A/ If it is regular the angles and sides would all be congruent. /B/ If it is concave lines drawn from the segments would pass through the polygon. /C/ Correct! /D/ Count the number of sides. 10. ANS: B Perimeter is the sum of the sides. TOP: Find the perimeters of polygons. KEY: Perimeter, Polygons NOT: /A/ What is the sum of the sides? /B/ Correct! /C/ Did you find the value of y? /D/ What is the value of y? 11. ANS: A Perimeter is the sum of the sides. TOP: Find the perimeters of polygons. KEY: Perimeter, Polygons NOT: /A/ Correct! /B/ How many sides does the figure have? /C/ Should all the sides have the same length? /D/ The sides are congruent. 12. ANS: C Angles are congruent only if their measures are equal. Point B may be closer to line AD or line DC so the measures would not be equal. TOP: Find counterexamples. KEY: Counterexamples NOT: /A/ What is the definition of congruent? /B/ What is the definition of congruent? /C/ Correct! /D/ Would that be a counterexample? 13. ANS: D Because m is squared in the example, m could be positive or negative. TOP: Find counterexamples. KEY: Counterexamples NOT: /A/ Subtract 6 from both sides. /B/ What about negative numbers? /C/ Subtract 6 from both sides. /D/ Correct! 14. ANS: C If two angles are supplementary their measures total 180. Either both are right or one is obtuse and the other acute. TOP: Find counterexamples. KEY: Counterexamples NOT: /A/ What is the definition of supplementary? /B/ What is the definition of supplementary? /C/ Correct! /D/ What is the definition of supplementary? 15. ANS: B Even though they have a common point, the two segments do not have to be on the same line. TOP: Find counterexamples. KEY: Counterexamples NOT: /A/ What is the definition of midpoint? /B/ Correct! /C/ What is the definition of midpoint? /D/ What is the definition of midpoint? 16. ANS: A Planes that intersect have a common line. TOP: Identify the relationships between two lines or two planes. KEY: Relationship Between Two Lines, Relationship Between Two Planes NOT: /A/ Correct! /B/ This plane has four lines to intersect with other planes. /C/ Do they all intersect CDI in a line? /D/ This plane has four lines to intersect with other planes. 17. ANS: B Coplanar segments that do not intersect are parallel. TOP: Identify the relationships between two lines or two planes. KEY: Relationship Between Two Lines, Relationship Between Two Planes NOT: /A/ Are those parallel to GF? /B/ Correct! /C/ Is that all? /D/ Is that all? 18. ANS: D Skew lines do not intersect and are not coplanar. TOP: Identify the relationships between two lines or two planes. KEY: Relationship Between Two Lines, Relationship Between Two Planes NOT: /A/ Are any of those segments in the same plane as segment BC? /B/ Skew lines are not coplanar. /C/ Do any of those segments intersect segment BC? /D/ Correct! 19. ANS: C Corresponding angles are congruent. Alternate interior angles are congruent. Consecutive interior angles are supplementary. Alternate exterior angles are congruent. TOP: Use algebra to find angle measures. KEY: Angles, Angle Measures NOT: /A/ What do supplementary angles add up to?/B/ What do the angles of a right triangle add up to? /C/ Correct! /D/ Is that triangle isosceles? 20. ANS: C Postulates and theorems: If corresponding angles are congruent, then lines are parallel. If given a line and a point not on the line, then there exists exactly one line through the point that is parallel to the given line. If alternate exterior angles are congruent, then lines are parallel. If consecutive interior angles are supplementary, then lines are parallel. If alternate interior angles are congruent, then lines are parallel. If 2 lines are perpendicular to the same line, then lines are parallel. TOP: Recognize angle conditions that occur with parallel lines. KEY: Angles, Parallel Lines NOT: /A/ What kind of angles are those? /B/ What kind of angles are those?/C/ Correct! /D/ Which lines are parallel? 21. ANS: A The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. TOP: Apply the Exterior Angle Theorem. KEY: Exterior Angle Theorem NOT: /A/ Correct! /B/ What is the sum of the measures of the angles in a triangle? /C/ Did you use the Exterior Angle Theorem?/D/ Use the Exterior Angle Theorem. 22. ANS: C The corresponding sides and angles can be determined from any congruence statement by following the order of the letters. TOP: Name and label corresponding parts of congruent triangles. KEY: Corresponding Parts, Congruent Triangles NOT: /A/ Did you follow the order of the letters?/B/ The corresponding sides and angles can be determined from any congruence statement by following the order of the letters. /C/ Correct! /D/ Did you follow the order of the letters? 23. ANS: D If each side of triangle PQR is the same length as the corresponding side of triangle STU, then the triangles are congruent. TOP: Use the SSS Postulate to test for triangle congruence. KEY: SSS Postulate, Congruent Triangles NOT: /A/ Use the SSS Postulate. /B/ Check your math. /C/ How do you determine if two triangles are congruent? /D/ Correct! 24. ANS: A TOP: Use properties of equilateral triangles. KEY: Equilateral Triangles NOT: /A/ Correct! /B/ What do you know about the sides of an equilateral triangle? /C/ How many degrees is each angle of an equilateral triangle? /D/ Did you add or subtract when solving for y? 25. ANS: A TOP: Use properties of equilateral triangles. KEY: Equilateral Triangles NOT: /A/ Correct! /B/ Did you set the sides equal to each other? /C/ How many degrees is each angle of an equilateral triangle? /D/ Check your math. 26. ANS: A If is an altitude, . Also, the measures of the angles of every triangle add up to 180. TOP: Identify and use altitudes in triangles. KEY: Altitudes, Triangles NOT: /A/ Correct! /B/ Which angle measures add up to 180?/C/ Which angles must measure 90°? /D/ Check your math. 27. ANS: C The measures of the angles opposite the sides given are compared. The larger the angle, the longer its side. TOP: Recognize and apply properties of inequalities to the relationships between angles and sides of a triangle. KEY: Properties of Inequality, Triangles NOT: /A/ Check the angles opposite the sides. /B/ Are the angles in the same or congruent triangles? /C/Correct!/D/ Check the angles opposite the sides. 28. ANS: D The sum of the lengths of any two sides must be greater than the third. TOP: Apply the Triangle Inequality Theorem. KEY: Triangles Inequality Theorem NOT: /A/ Did you check all the sums? /B/ Add two sides and compare to the third. /C/ Add two sides and compare to the third./D/ Correct! 29. ANS: A The sum of the lengths of any two sides must be greater than the third. TOP: Determine the shortest distance between a point and a line. KEY: Distance, Distance Between a Point and a Line NOT: /A/ Correct! /B/ Would both sides have to be longer than the base?/C/ Is the sum of the two sides longer than the base? /D/ Is that the shortest possible length? 30. ANS: B Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional. TOP: Identify similar figures. KEY: Similar Figures NOT: /A/ Check the size of the angles./B/ Correct! /C/ Determine the ratio of the corresponding sides. /D/ Check the similarity statement. 31. ANS: A Determine the ratio of corresponding parts. Use the ratio to find the missing information. TOP: Use similar triangles to solve problems. KEY: Similar Triangles, Solve Problems NOT: /A/ Correct! /B/ Check your ratio. /C/ What are the values of AB and BC? /D/ Check your addition. 32. ANS: A Determine the ratio of corresponding parts. Use the ratio to find the missing information. TOP: Use similar triangles to solve problems. KEY: Similar Triangles, Solve Problems NOT: /A/ Correct! /B/ Check your addition. /C/ Check the ratios. /D/ Check the ratios and your multiplication. 33. ANS: A Use the Proportional Perimeter Theorem to find the perimeter of the selected triangle. TOP: Recognize and use proportional relationships of corresponding perimeters and altitudes. KEY: Corresponding Perimeters, Corresponding Altitudes NOT: /A/ Correct! /B/ This is the perimeter of the wrong triangle. /C/ This is the difference between the perimeters of the two similar triangles. /D/ Remember to find all sides of the triangle. 34. ANS: D Write a proportion. Substitute the given values for the sides. Cross multiply, and simplify. TOP: Recognize and use proportional relationships of corresponding angle bisectors and medians of similar triangles. KEY: Corresponding Angle Bisectors, Corresponding Medians NOT: /A/ This is the value of the variable. /B/ Which side does the question ask for? /C/ This is the ratio of the corresponding parts. /D/ Correct! 35. ANS: A Draw a picture of the situation. Determine which trigonometric ratio should be used to solve. Substitute the numbers given. Solve for the answer. TOP: Solve problems using trigonometric ratios. KEY: Trigonometric Ratios, Solve Problems NOT: /A/ Correct! /B/ Check the trigonometric ratio. /C/ Which trigonometric ratio should be used? /D/ Which trigonometric ratio should be used? 36. ANS: B Draw a picture of the situation. Determine which trigonometric ratio should be used to solve. Substitute the numbers given. Solve for the answer. TOP: Solve problems using trigonometric ratios. KEY: Trigonometric Ratios, Solve Problems NOT: /A/ Remember to include the initial height of one kilometer. /B/ Correct! /C/ Which trigonometric ratio should be used? /D/ Which trigonometric ratio should be used? 37. ANS: A Draw a picture of the situation. Determine which trigonometric ratio should be used to solve. Substitute the numbers given. Solve for the answer. TOP: Solve problems using trigonometric ratios. KEY: Trigonometric Ratios, Solve Problems NOT: /A/ Correct! /B/ Do not use the sine ratio. /C/ Do not use the tangent ratio. /D/ What is the cosine ratio? 38. ANS: C Draw a picture of the situation. Determine which trigonometric ratio should be used to solve. Substitute the numbers given. Solve for the answer. TOP: Solve problems involving angles of elevation. KEY: Angle of Elevation NOT: /A/ Did you use the inverse sine to solve. /B/ Do not use the tangent ratio. /C/ Correct! /D/ Do not use the cosine ratio. 39. ANS: A Draw a picture of the situation. Determine which trigonometric ratio should be used to solve. Substitute the numbers given. Solve for the answer. TOP: Solve problems involving angles of depression. KEY: Angle of Depression NOT: /A/ Correct! /B/ This is the horizontal distance from one horse to the hang glider. /C/ This is the horizontal distance from one horse to the hang glider. /D/ Subtract rather than add the distances. 40. ANS: A Substitute the given values into the Law of Cosines. Simplify the equation. Find the measure of the stated angle by using inverse cosine. TOP: Solve problems by using the Law of Cosines. KEY: Law of Cosines, Solve Problems NOT: /A/ Correct! /B/ Which angle goes with each vertex? /C/ Which angle goes with each vertex? /D/ The triangle is not equilateral. 41. ANS: A Locate the angle on the parallelogram. Using the properties of parallelograms, determine which angle is congruent to that angle. TOP: Recognize and apply properties of the sides and angles of parallelograms. KEY: Parallelograms, Properties of Parallelograms NOT: /A/ Why are these angles congruent? /B/ Check the angle and reason. /C/ Correct! /D/ This angle is not congruent to the original angle. 42. ANS: A Using the method indicated, determine if the points form a parallelogram. If the opposite sides are congruent, the slopes of opposite sides are congruent, or the diagonals share the same midpoint, then the points form a parallelogram. TOP: Prove that a set of points forms a parallelogram in the coordinate plane. KEY: Parallelograms, Determining a Parallelogram NOT: /A/ Correct! /B/ Which method was used to solve the problem?/C/ Which method was used to solve the problem? /D/ This is not valid reason for the quadrilateral to be a parallelogram. 43. ANS: B All sides of a rhombus are congruent. TOP: Recognize and apply the properties of rhombi. KEY: Rhombi, Properties of Rhombi NOT: /A/ All sides of a rhombus are congruent. /B/ Correct! /C/ All sides of a rhombus are congruent. /D/ Do not use properties of special triangles. 44. ANS: A Plot the vertices on a coordinate plane. Determine if the diagonals are perpendicular. If so, the quadrilateral is either a rhombus or square. Use the distance formula to compare the lengths of the diagonals. If the diagonals are congruent and perpendicular, the quadrilateral is a square. TOP: Recognize and apply the properties of squares. KEY: Squares, Properties of Squares NOT: /A/ Correct! /B/ Are the angles congruent? /C/ Remember to list all that apply./D/ Are the sides congruent? 45. ANS: D To find the other base, substitute the given values into the formula . TOP: Recognize and apply the properties of trapezoids. KEY: Trapezoids, Properties of Trapezoids NOT: /A/ Do not subtract the base from the median. /B/ AB is the median not a base. /C/ Do not add the median and base./D/ Correct! 46. ANS: A The circumference formula is Diameter . The diameter shown also happens to be the diagonal of a square, so it can be found by multiplying the side of the square by . So the diameter would be side and the circumference would be side TOP: Solve problems involving the circumference of a circle. KEY: Circles, Circumference NOT: /A/ Correct! /B/ Check your diameter again. /C/ You need a radical 2 in your answer./D/ You need to double your radius. 47. ANS: B is complementary with , so its measure is 90 – 50 = 40. TOP: Recognize major arcs¸ minor arcs¸ semicircles¸ and central angles and their measures. KEY: Major Arcs, Minor Arcs, Semicircles, Central Angles NOT: /A/ Did you subtract carefully? /B/ Correct! /C/ With what angle is it complementary? /D/ What is the measure of angle BAE? 48. ANS: B Since and are congruent and perpendicular to separate chords, the chords and must also be congruent. Additionally, measure of and bisect these chords, so TS = RS. So take the given and double it. That will be the measure of and also . TOP: Recognize and use relationships between arcs and chords and diameters. KEY: Arcs, Chords, Diameters NOT: /A/ What is the relationship between PR and TS? /B/ Correct! /C/ Isn't PR longer than TS? /D/ What is the relationship between PR and TS? 49. ANS: B The measure of arc BC is 2 . Since the full circle measures 360, then m arc AD is 360 – ( + + ). Finally, since is an inscribed angle for arc , its measure is half m arc AD. TOP: Find measures of inscribed angles. KEY: Inscribed Angles, Measure of Inscribed Angles NOT: /A/ Can you find the measure of arc AD?/B/ Correct! /C/ Inscribed angles are half the measure of the arc. /D/ Focus on arc AD. 50. ANS: B The triangle shown is a right triangle since the tangent segment, FE, intersects a radius, DE, which always results in a right angle. So to solve for x, use the Pythagorean Theorem. Note that since they are both radii of the same circle. The equation from the Pythagorean Theorem is . TOP: Use properties of tangents. KEY: Tangents NOT: /A/ Use the Pythagorean Theorem and mDE = x./B/ Correct! /C/ Is the triangle a right? /D/ Use the Pythagorean Theorem and mDE = x. 51. ANS: C When two secants intersect in the interior of a circle, then the measure of an angle formed by this intersection is equal to one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle. In this diagram, the measure of one of the intercepted arcs for is not given, but it can be found since the sum of all of the arcs must be 360. Subtracting the sum of the other 3 arcs from 360, leaves 110. That means . TOP: Find measures of angles formed by lines intersecting on or inside a circle. KEY: Measure of Angles, Circles NOT: /A/ Did you use the correct arcs? /B/ Add the intercepted arcs and divide by 2. /C/ Correct! /D/ Did you find the measure of the unlabeled arc? 52. ANS: D When two secants intersect in the exterior of a circle, then the measure of the angle formed is equal to one-half the positive difference of the measures of the intercepted arcs. Here that gives the equation . Solving this equation results in the answer x = 10. TOP: Find measures of angles formed by lines intersecting outside the circle. KEY: Measure of Angles, Circles NOT: /A/ How are the angle and the intercepted arcs related? /B/ Check your subtraction. /C/ Did you subtract carefully? /D/ Correct! 53. ANS: C When a secant and tangent intersect in the exterior of a circle, then the measure of the angle formed is equal to one-half the positive difference of the measures of the intercepted arcs. Here that gives the equation . Solving this equation results in an answer of x = 25. Note that the intercepted arc nearest the angle is 40 since 360 – 230 – 90 = 40. TOP: Find measures of angles formed by lines intersecting outside the circle. KEY: Measure of Angles, Circles NOT: /A/ Check your subtraction. /B/ Did you subtract carefully? /C/ Correct. /D/ Check your subtraction. 54. ANS: C The products of the segments for each intersecting chord are equal, so 2 x = 3 6. Solving this equation, x = 9. TOP: Find measures of segments that intersect in the interior of a circle. KEY: Circles, Interior of Circles NOT: /A/ Use multiplication, not addition. /B/ Multiply the segments and set them equal to each other. /C/ Correct! /D/ Multiply the segments and set them equal to each other. 55. ANS: D When a secant segment and a tangent segment intersect in the exterior of a circle, set the product of each external part of the secant segment and the entire secant segment equal to the square of the tangent segment. Here that gives the equation x x = 2 (2 + 6). Solving this equation results in x = 4. TOP: Find measures of segments that intersect in the exterior of a circle. KEY: Circles, Exterior of Circles NOT: /A/ Check your multiplication. /B/ You need to multiply, not add. /C/ Check the segments in your multiplication. /D/ Correct! 56. ANS: A When two secant segments intersect in the exterior of a circle, set an equality between the product of each external segment and the entire segment. Here that gives the equation 2 12 = x (x + 5). Solving this equation results in x = –8, 3. Since x is a measurement and cannot be negative, the answer is x = 3. TOP: Find measures of segments that intersect in the exterior of a circle. KEY: Circles, Exterior of Circles NOT: /A/ Correct! /B/ Check your multiplication. /C/ You need to multiply, not add. /D/ Check the segments in your multiplication. 57. ANS: A The perimeter is the sum of all four sides. So add the two sides given and then double the result. This accounts for the other two sides which are the same as those given. For example, if the two given sides are 8 and 12, the perimeter is 2 (8 + 12) = 40. The area is base height. The base is given. The height can be found by recognizing the 30°-60°-90° triangle at the left. The height is the long leg which is found by taking the hypotenuse (the given slanted side) and dividing by 2, then multiplying by the square root of three. For example, if the slanted side is 8, the height is 82 . Multiply that height times the given base and you have the area. TOP: Find areas of parallelograms. KEY: Parallelograms, Area, Area of Parallelograms NOT: /A/ Correct! /B/ Perimeter is the sum of all four sides. Area is base times height./C/ Perimeter is the sum of all four sides. /D/ Area is base times height. 58. ANS: A The area of a rhombus is found by the formula diagonals of the rhombus are 20 and 30, the area is . For example, if the 20 30 = 300. TOP: Find areas of rhombi. KEY: Area, Rhombi, Area of Rhombi NOT: /A/ Correct! /B/ Area is diagonal 1 times diagonal 2 divided by 2./C/ How do you find the area of a rhombus? /D/ What is the formula for the area of a rhombus? 59. ANS: D This figure represents a rectangle attached to a right triangle. The area of the rectangle is . The area of the triangle is . So the total area of the figure is . TOP: Find areas of irregular figures. KEY: Area, Irregular Figures, Area of Irregular Figures NOT: /A/ The dimensions of the rectangle are 8 by 13. /B/ The area of the rectangle is 0.5 * 6 * 6. /C/ The dimensions of the rectangle are 8 by 13. /D/ Correct! 60. ANS: D This figure represents a rectangle with two identical semicircles cut out on each side. The semicircles combine to make one circle with a diameter of 24 and a radius of 12. The area of the rectangle is . The area of the circle is , using 3.14 for . So the area of the figure is . TOP: Find areas of irregular figures. KEY: Area, Irregular Figures, Area of Irregular Figures NOT: /A/ The radius of the circle is 12, not 24. /B/ The two semicircles add up to one circle with a radius of 12. /C/ The two semicircles add up to one circle with a radius of 12. /D/ Correct! 61. ANS: C The formula for the lateral area of a regular pyramid is where P is the perimeter of the base and is the slant height of a face of the pyramid. For example, if the slant height is given as 12 and the side as 14, then the perimeter is 56 (4 time 14, since it is a square), and the lateral area is units2. The lateral area of the cube is perimeter times height. The perimeter in this example as 56, and the height is 14 since it is a cube. So the lateral area of the cube is . Finally, the total lateral area of the solid is the sum of the lateral area of the pyramid and the lateral area of the cube. Add the two previous answers to get units2. TOP: Find lateral areas of regular pyramids. KEY: Lateral Area, Pyramids, Lateral Area of Pyramids NOT: /A/ The lateral area is the sum of the lateral areas of the pyramid and the cube. /B/ You found the surface area, not the lateral area. /C/ Correct! /D/ The lateral area is the sum of the lateral areas of the pyramid and the cube. 62. ANS: A The formula for the surface area of a regular pyramid is + B where P is the perimeter of the base, is the slant height and B is the area of the base of the pyramid. Since this is a regular (or equilateral) triangle, to find the perimeter of the base, multiply the length of one side by 3. To find the area of the triangular base, the formula is . To find the length of the slant height, use the Pythagorean Theorem as follows: . For example, if the lateral edge is given as 16 and the side as 18, then the slant height is surface area is . The units2. TOP: Find surface area of regular pyramids. KEY: Surface Area, Pyramids, Surface Area of Pyramids NOT: /A/ Correct! /B/ What is the formula for finding the surface area of a regular pyramid? /C/ The surface area of a regular pyramid is one-half of the perimeter of the base times the slant height + the area of the base./D/ You only found the lateral area. You need the surface area. 63. ANS: B The formula for the lateral area of a cone is r where r is the radius of the circular base and is the slant height of the cone. For example, if the slant height is given as 10 and the radius as 6, then the lateral area is then computed as units2. TOP: Find lateral areas of cones. KEY: Lateral Area, Cones, Lateral Area of Cones NOT: /A/ What is the formula for the lateral area of a cone? /B/ Correct! /C/ You found the surface area. You need to find the lateral area. /D/ The lateral area of a cone is pi times radius times slant height. 64. ANS: C A plane that intersects a sphere through its center will result in a great circle by definition. TOP: Recognize and define basic properties of spheres. KEY: Spheres NOT: /A/ The intersection will result in a great circle. /B/ The intersection will result in a great circle. /C/ Correct! /D/ The intersection will result in a great circle. 65. ANS: A The radii of a sphere, like those of a circle, are all the same length. So the statement is true. TOP: Recognize and define basic properties of spheres. KEY: Spheres NOT: /A/ Correct! /B/ All radii of the same sphere are the same length. /C/ All radii of the same sphere are the same length. /D/ This is not a relevant counterexample. 66. ANS: A The surface area of a sphere is found by the formula , where r is the radius of the sphere. This diagram gives the diameter, so begin by computing the radius which is half the diameter. For example, if the diameter is given as 12, then the radius is 6 and the surface area would be . TOP: Find surface area of spheres. KEY: Surface Area, Spheres, Surface Area of Spheres NOT: /A/ Correct! /B/ Does the formula indicate cubing the radius?/C/ Did you use the correct formula? /D/ Did you leave pi out of the formula? 67. ANS: D The surface area of a sphere is found by the formula , where r is the radius of the sphere. This problem gives the circumference, so begin by computing the radius which can be found by the formula . For example, if the circumference is given as 12.56, then the radius is the surface area would be surface area is and . If the solid is a hemisphere, then the formula for the , since a hemisphere has half the surface area of the sphere plus the area of the great circle on its base . TOP: Find surface area of spheres. KEY: Surface Area, Spheres, Surface Area of Spheres NOT: /A/ Does the formula indicate cubing the radius?/B/ Did you leave pi out of the formula? /C/ Did you use the correct formula? /D/ Correct! 68. ANS: C The volume of the cone is cm3. The volume of the ice (a sphere) is cm3. So the volume of the ice is greater than the volume of the cone, causing it to overflow the cone. TOP: Solve problems involving volumes of spheres. KEY: Volume, Spheres, Volume of Spheres NOT: /A/ Check your volume calculations. /B/ Check your volume calculations. /C/ Correct! /D/ This problem can be solved with the information given. 69. ANS: A The formula for the volume of a sphere is . So for this sphere, the volume is cm3. TOP: Solve problems involving volumes of spheres. KEY: Volume, Spheres, Volume of Spheres NOT: /A/ Correct! /B/ The formula has a 4/3, not a 1/3. /C/ You have to cube the radius. /D/ The formula has a 4/3, not a 1/3. You also need to cube the radius. 70. ANS: B Two spheres that are congruent are the same size and shape. Therefore, they would have the same surface area. So this statement is always true. TOP: State the properties of similar solids. KEY: Similar Solids NOT: /A/ Congruent means same size and shape. /B/ Correct! /C/ Congruent means same size and shape.