Honors Geometry 2015-2016
The following formulas will be provided in the student examination booklet. Pythagorean Theorem In right triangle with right angle at point : 2 2 2 a b c b c a Trigonometry In a right triangle with acute angle : side opposite sin hypotenuse side adjacent to cos hypotenuse side opposite tan side adjacent to In any triangle : sin sin sin a b c or The Law of Sines a b c sin sin sin 2 2 2 c a b 2abcos The Law of osines rea of a triangle = 1 sin 2 ab MPS
Items on this review are grouped by Unit/Topic Unit 1, Topic 1 1. Write the three undefined terms in geometry. 2. onstruct the angle bisector of angle below. 3. onstruct a right angle at point below. MPS Review Page 1
4. road construction crew wishes to construct a new road parallel to the current road, and passing through the town of Herkimer, marked by point H below. onstruct the location of the road. current road H 5. The perpendicular bisector of is constructed. What is true about every point on that perpendicular bisector? 6. The angle bisector of E is constructed. What is true about every point on the angle bisector? 7. line is constructed parallel to a given line. What is true about the corresponding points on the lines? MPS Review Page 2
Unit 1, Topic 2 8. Point 1, 4 is to be transformed to point using the translation rule xy, x 3, y 6. What are the coordinates of point? 9. Triangle has been transformed to triangle. y O x a. State in words, the transformation(s) that produce triangle. b. Write a function rule that represents this transformation. c. Why must? d. On the coordinate plane above, sketch the reflection of across the x-axis. Label the triangle EF. Write the function rule for this transformation. MPS Review Page 3
10. Let x, y be a point on the coordinate plane. Write the coordinates of the image point if x, y undergoes the following transformations. a. Reflected across the x-axis. b. Reflected across the y-axis. c, Translated six units to the right and two units downward. d. Rotated clockwise 90 degrees about the origin. e. Rotated 180 degrees about the origin. f. Rotated counter-clockwise 90 degrees about the origin. g. Translated two units down, then reflected about the x-axis. h. Reflected across the line x 1. 11. y O x a. Reflect the figure above across the line y 1. Label the image. b. oes this transformation preserve lengths and angle measurements? Justify your answer. c. If the figure was reflected across the x-axis, then translated two units upward, would the result be the same as the transformation in part a)? MPS Review Page 4
12. Write the three rigid transformations. 13. If a figure undergoes a rigid transformation, then the transformed figure must be to the original figure. 14. Look at the figure below. y O x omplete each statement such that each transformation will map the figure onto itself. a. reflection across the axis or the line x. b. rotation of degrees about the point. MPS Review Page 5
15. Triangle undergoes a transformation to produce triangle. The triangles are shown below. y O x a. an the transformation above be performed using a single reflection? Justify your answer. b. an the transformation above be performed by a single translation? Justify your answer. c. etermine a transformation with three reflections that will produce triangle. MPS Review Page 6
16. The line y 3x 2 is graphed on the coordinate plane below. y O x Write the equation of the line if y 3x 2 is a. reflected across the x-axis. b. reflected across the y-axis. c. translated three units to the right and one unit up. MPS Review Page 7
Unit 1, Topic 3 17. If figure is congruent to figure EFGH, complete the congruence statements. 18. If all corresponding sides of two figures are congruent, and all corresponding angles of the two figures are congruent, then the figures must be. Items 19 through 23 use the figure below, where point S is the midpoint of WU. Therefore WS US. R T W S U You wish to prove WSR UST. 19. To prove this congruence by SSS, what two congruence statements are needed in addition to WS US? 20. To prove this congruence by SS, what two congruence statements are needed in addition to WS US? 21. To prove this congruence by S, what two congruence statements are needed in addition to WS US? 22. To prove this congruence by S, what two congruence statements are needed in addition to WS US? 23. If RW TU and R T, must RWS be congruent to TUS? Justify your answer. MPS Review Page 8
24. onsuela wants to determine the length of a power line that will be stretched over a lake. She cannot walk through the lake. She was able to take some measurements, hoping to determine the length of the power line. Her measurements are shown below. 625 ft 500 ft 450 ft Figure NOT drawn to scale 450 ft 500 ft Power Line onsuela believes that the length of the power line is 625 feet, but she s not sure how to explain this to her boss. Using what you know about triangle congruence, help onsuela by writing a brief report as to why the length of the power line is 625 feet. MPS Review Page 9
In items 25 through 30 below, information is given about triangles and FE. State whether the triangles can be proven congruent by S, SSS, S, or SS. If the triangles cannot be proven congruent, state why. E 25. FE, F, F. F 26. F, E,. E F 27. F, E, FE. E F 28. FE, F, E. E F 29. E, F, E. E F E 30. F, FE, F MPS Review Page 10
Unit 1, Topic 4 31. Every quadrilateral that is a parallelogram has certain properties. List all these properties. 32. What properties does a rectangle have in addition to those of a parallelogram? 33. What properties does a square have in addition to those of a rectangle? 34. Triangle EF is isosceles with E. Which sides are congruent? 35. In the figure to the right, Figure NOT drawn to scale is the midpoint of and E is the midpoint of. 8, 22 The ratio of to is 2 to 3. etermine the lengths of E and E. E MPS Review Page 11
36. In the figure below, is the midpoint of and E is the midpoint of. E omplete the statements using the items in the box on the right. Each item may be used more than once or not at all. a. b. E E E E 2:1 1:2 c. is parallel to d. E is one-half the length of e. The ratio : is f. The ratio E: is MPS Review Page 12
37. If two parallel lines are cut by a transversal: a. Which angle pairs are congruent? b. Which angle pairs are supplementary? 38. Given: m n, 1 16 Prove: p q 1 3 2 4 p 5 6 7 8 q m 9 10 11 12 15 13 14 16 n 39. Given: E E Prove: E MPS Review Page 13
40. Given: is the midpoint of E. E, E Prove: E 41. elow is a figure that will help you prove that the sum of the angles of a triangle is 180 degrees. The two lines are parallel. 1 2 3 m 4 5 n Write a proof for this theorem. MPS Review Page 14
Unit 2, Topic 1 Figure NOT drawn to scale 42. In the figure to the right, E. E 2, E 8, 9, 18 a. Prove that E ~. E b. etermine the length of. c. etermine the length of E. MPS Review Page 15
43. Jory wants to measure the length, L, of a pond. The figure below shows the measurements she will use to determine the length of the pond. In the figure below, E and intersect at point. 30 ft Figure NOT drawn to scale 20 ft 25 ft 50 ft 40 ft L E a. Which similarity postulate/theorem may be used to prove E? b. What is the length, L, of the pond? Show how you determined your answer. MPS Review Page 16
44. If two figures are similar, then a. orresponding sides are (proportional/congruent). b. orresponding angles are (proportional/congruent). 45. In the figure below, QRS ~ QTU. Q S U R T a. Prove that RS TU. b. omplete the proportions. QS SU QR RT RT RS MPS Review Page 17
46. The figure below shows. will be the image of after a dilation with center and scale factor 5. Item ank 5 1 5 omplete the statements using the item bank on the right. a. The point will be the same point as point. parallel to perpendicular to b. will be. c. The perimeter of will be times the perimeter of. MPS Review Page 18
47. On the graph below, quadrilateral has been dilated, with the center of dilation the origin, then translated down six units, to create quadrilateral. O -10-9 - 8-7 - 6-5 - 4-3 - 2-1 1 2 3 4 5 6 7 8 9 10-1 -2-3 -4-5 -6-7 -8-9 -10 y 10 9 8 7 6 5 4 3 2 1 x a. What is the scale factor of the dilation? b. What is the ratio of the length of to the length of? c. How are the measures of and related? MPS Review Page 19
48. Triangle is shown on the coordinate grid below. 10 9 8 7 6 5 4 3 2 1-10 - 9-8 - 7-6 - 5-4 - 3-2 - 1 O 1 2 3 4 5 6 7 8 9 10-1 P y -2-3 -4-5 -6-7 -8-9 -10 x Triangle is to be dilated by a scale factor of three with the center of dilation the. point P 5, 6 to produce triangle a. On the coordinate plane above, sketch. b. What is the ratio of the perimeter of to the perimeter of? c. Is there a sequence of rigid transformations that will produce the same result? Justify your answer. MPS Review Page 20
49. Quadrilateral has been dilated. The dilation results in the quadrilateral. O -10-9 - 8-7 - 6-5 - 4-3 - 2-1 1 2 3 4 5 6 7 8 9 10-1 -2-3 -4-5 -6-7 -8-9 -10 y 10 9 8 7 6 5 4 3 2 1 x a. What is the scale factor of the dilation? b. What point is the center of dilation? MPS Review Page 21
50. Pentagons E and FGHIJ are similar. E y 10 15 J y 1.5 I x F H 12 G Figure NOT drawn to scale a. What is the value of x? b. What is the value of y? MPS Review Page 22
51. Jack wishes to find the height of a tree. He puts a stick into the ground and uses the tree s shadow to take some measurements. The figure below shows his measurements. Figure NOT drawn to scale h ft 5 ft 135 ft 9 ft a. etermine the height of the tree. b. etermine the distance from the tip of the tree s shadow to the top of the tree. MPS Review Page 23
52. raw examples of two triangles that are similar by each similarity postulate/theorem. a. raw examples of two triangles that are similar by SSS Similarity. b. raw examples of two triangles that are similar by Similarity. c. raw examples of two triangles that are similar by SS Similarity. MPS Review Page 24
Unit 2, Topic 2 53. Given right, use the item bank to identify the following. Items may be used more than once. Item ank a. Leg opposite h. Tangent ratio of b. Leg opposite i. The angle whose Sine ratio is c. Sine ratio of j. The angle whose Tangent Ratio is d. osine ratio of k. Hypotenuse e. Tangent ratio of f. Sine ratio of g. osine ratio of MPS Review Page 25
54. Let be a right triangle, with right angle at. Which statements are true? heck all that apply. and are complementary m 90 o m sin tan cos 2 2 2 sin cos cos sin 90 o If, then m 45 o 55. Look at the two triangles below. 16 12 16 20 20 12 Saundra says that since the two triangles are congruent that sin sin. Is Saundra correct? Justify your answer. MPS Review Page 26
56. Look at the figure below. Figure NOT drawn to scale Q 12 R 8 S 15 6 10 T 15 U a. Show that the value of cos S is the same if SQU or SRT is used. b. What is the value of sinu? c. Is cosu cost? Why or why not? MPS Review Page 27
57. skateboarder has a piece of plywood that she wants to use as a ramp. The plywood is 16 feet long. She wants to put a vertical support under the ramp so that the ramp is at a 20 o angle to the ground. This is shown in the figure below. The view is from the side of the ramp. 16 ft. ramp Figure NOT drawn to scale Vertical support 20 o How high is the vertical support? Round your answer to the nearest hundredth of a foot. 58. Josh has a 17 ft. ladder. He would like to place the ladder against the side of a vertical wall so that the top of the ladder is 15 feet up the wall. Safety guidelines state that the angle that the ladder makes with the horizontal can be no more than 70 degrees. a. an Josh safely place the ladder against the wall? b. Suppose that Josh wanted the top of the ladder to be 24 feet up the wall. In order to meet the safety guidelines, what is the minimum length of ladder that Josh needs? Round your answer to the nearest hundredth of a foot. MPS Review Page 28
59. plane takes off from a runway and climbs at an angle of elevation of 15 o. a. fter the plane has travelled one mile (5280 feet) how far has the plane travelled horizontally? Round your answer to the nearest foot. b. fter the plane has travelled one mile (5280 feet) how far has the plane travelled vertically? Round your answer to the nearest foot. 60. farmer wishes to make separate sections of a piece of land as shown in the figure below. ll triangles are right triangles. 65 m Figure NOT drawn to scale 20 m x y 15 m a. What is the value of x? b. What is the value of y? c. What is the area of the trapezoid? MPS Review Page 29
61. swimmer tries to swim straight across a stream. The stream is 100 feet wide. current in the stream, perpendicular to his intended path, caused him to land 20 feet downstream from his intended landing point. 20 ft. Figure NOT drawn to scale Intended path 100 ft ctual path a. How far did the he swim? Give your answer to the nearest tenth of a foot. b. t what angle,, was his actual path to the intended path? Give your answer to the nearest tenth of a degree. MPS Review Page 30
62. Kevin purchased two farms. urrently, the two farms are connected by two roads. Kevin wants to construct a new road connecting the two farms directly. The figure below represents this situation. Figure NOT drawn to scale Farm New road Farm 14 miles 23 miles 68 o a. What will be the length of the new road? Your answer should be correct to the nearest hundredth of a mile. b. The Federal Government is planning on making the area between the two existing roads and the new road into a national park. What is the area of the park? Your answer should be correct to the nearest hundredth of a square mile. MPS Review Page 31
63. roof is being constructed on a building and slopes differently on the two sides, as shown in the figure below. Figure NOT drawn to scale 79 o 43 o 20 ft What is the length of the roof represented by? You answer should be correct to the nearest hundredth of a foot. MPS Review Page 32