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Name ______________________________________________ Date _______ End of Module 2 Review Sheet (2.1 β 2.28) CC Geometry 1. In the diagram below, ππΜ Μ Μ Μ and ππ Μ Μ Μ Μ intersect at T, ππΜ Μ Μ Μ is drawn, and ππΜ Μ Μ Μ || ππ Μ Μ Μ Μ . Prove that βπππ ~ βπ ππ.
2. Simplify each of the following:
A) yx275 B) 87285 yx C) 42 452 cddcοΏ½
3. Are the triangles shown below similar? Explain. If the triangles are similar, write a similarity statement. A) B) C) D) E)
Statement Reason SQ and PR intersect at T, PQ Given Is drawn and PS ll QR <STP = <QTR Vertical angles are congruent <SPT = <QRT If parallel lines are cut by a transversal then alternate interior angles are congruent PST = RQT AA=AA
Answer key
9. The scale factor between two triangles is 2. If the area of the larger triangle is 36 cm2, find the area of the smaller triangle.
10. Given βπΌπ½πΎ~βπΏππ, find the length of πΌοΏ½Μ οΏ½ and the length of πΌπΎΜ Μ Μ .
11. Given βπΎπ·πΆ~βπΎπΏπ, solve for x.
12. Sam wants to enlarge a triangle with sides 3, 6 and 6 inches. If the shortest side of the new triangle is 13
inches, how long will the other two sides be? 13. Given βπΎπ·πΆ~βπΎπΏπ, solve for x.
14. Sam wants to enlarge a triangle with sides 3, 6 and 6 inches. If the shortest side of the new triangle is 13
inches, how long will the other two sides be?
Repeat question
Repeat question
15. Given βπ½πΎπΏ~βπΊπΎπ», solve for x. 16. Given βπ΄π΅πΆ~βπ΄π·πΈ, find the length of CE and BD. 17. Given βπ΄π΅πΆ~βπππ, solve for x and y.
18. Given βπ΄π΅πΆ~βπππ, solve for x and y.
19. Find the center of dilation for the following:
20. In the following diagram, 65
''CBDAreaABDArea . If CD = 12, find: (a) AD (b) AC
21. Triangle ABC has a right angle at A and AD is the altitude from A to hypotenuse BC. Solve for x.
A) B)
22. The diagram below shows βπ΄π΅πΆ, with π΄πΈπ΅Μ Μ Μ Μ Μ Μ , π΄π·πΆΜ Μ Μ Μ Μ Μ , and . Prove that βπ΄π΅πΆ is similar to βπ΄π·πΈ.
23. In the diagram below, , , , and . Prove that .
Statement Reason <ACB=<AED Given <A=<A Reflexive Property ABC ADE AA=AA
Statement Reason BFCE, AB l BE, DE l BE. Given <BFD=<ECA <B and <E are right angles Perpendicular lines form right angles <B =<E All right angles are congruent <BFD + <DFE = 180 Linear pairs form supplementary <ECA + <ACB = 180 angles <BFD+<DFE =<ECA+ Substitution <ACB <DFE=<ACB Subtraction Postulate ABC DEF AA=AA
24. Dean constructed the composition of dilations shown below. Triangle AβBβCβ is 1/2 times the size of triangle ABC, and triangle AβBβCβ is 5
2 the size of triangle AβBβCβ.
a) Determine the scale factor from βπ΄π΅πΆ to βπ΄"π΅"πΆ".
b) Find the center of dilation mapping βπ΄π΅πΆ to βπ΄β²π΅β²πΆ and label O1.
c) Find the center of dilation mapping βπ΄β²π΅β²πΆβ² to βπ΄"π΅"πΆ" and label O2.
d) What is the relationship between the centers of dilations found is parts (b) through (c)?
25. A similarity transformation for triangle π΄π΅πΆ is described by (π·π,2 (ππ·πΊ β‘ (βπ΄π΅πΆ))) . Locate and label the
image of triangle π΄π΅πΆ under the similarity.
26. In the diagram below, π΄π΅Μ Μ Μ Μ || π·πΈΜ Μ Μ Μ . If AB = 20 inches, DE = 12 inches, and BC = 15 inches, what is the length of π·πΆΜ Μ Μ Μ ?
27. The scale factor between two triangles is 5. If the area of the smaller triangle is 20 in2, find the area of the
larger triangle. 28. Triangle ABC has a right angle at A and AD is the altitude from A to hypotenuse BC. Solve for x.
29. In the diagram to the right, π΄πΆ = 35. If the ratio of the areas of the triangles is Area βπ΄π΅πArea βπΆπ΅π
= 25, find x and y.
30. Find the missing length indicated for the diagrams below.
A) B) C)
31. Triangle ABC has a right angle at A and AD is the altitude from A to hypotenuse BC. Solve for x. A) B)
32. Solve for the missing side lengths and express your answers in simplest radical form.
A) B) C) D) E) F)
33. Find the value of each trig ratio
A) Tan Z B) Cos C C) Sin C
34. Find the measure of the indicated angle to the nearest tenth of a degree. A) B) C)
35. Find the missing side to the nearest hundredth.
A) B) C) D) 36. In the accompanying diagram, x represents the length of a ladder that is leaning against a wall of a
building, and y represents the distance from the foot of the ladder to the base of the wall. The ladder makes a 60Β° angle with the ground and reaches a point on the wall 17 feet above the ground. Find, to the nearest foot, the measure of x and y.
37. A 10-foot ladder is placed against the side of a building as shown in figure 1 below. The bottom of the ladder is 8 feet from the base of the building. In order to increase the reach of the ladder against the building, it is moved 4 feet closer to the base of the building as shown in figure 2. To the nearest foot, how much further up the building does the ladder now reach?
38. A cable 20 feet long connects the top of a flagpole to a point on the ground that is 16 feet from the base of the pole. How tall is the flagpole?
39. Find the value of x that makes each statement true.
a. Sin (x + 2) = Cos (2x β 11) d. Sin (x) = Cos 54
40. Calculate the missing side length, CB, to the nearest hundredth.
41. Calculate the missing side length, CB, to the nearest hundredth.
42. If cos π = 45, find sinT and tanT in simplest radical form if necessary.
43. If sin π = 3
4, find cosT and tanT in simplest radical form if necessary.
44. If tan π = 10, find sinT and cosT in simplest radical form if necessary.
45. If cos π = 22 , find sinT and tanT in simplest radical form if necessary.
46. If sin π = 66 , find cosT and tanT in simplest radical form if necessary.
47. a. State whether each triangle should be solved by using the Law of Sines or the Law of Cosines. b. Find the value of x in each triangle. Round your answers to the nearest tenth. A) B) C)
48. Triangle MNO has adjacent sides with length 12 and 17 and an included angle of 41o. Find the area of the
triangle to the nearest hundredth.
49. Find to the nearest integer the area of an isosceles triangle if the measure of the vertex angle is q42 and the measure of each of the congruent sides is 12.
50. In the accompanying diagram, the base of a 15-foot ladder rests on the ground 4 feet from a 6-foot fence.
a.) If the ladder touches the top of the fence, and the side of the building, what angle to the nearest degree does the ladder make with the ground?
b.) Using your answer from part a, find how high to the nearest tenth of a foot the ladder reaches up the side of the building.
51. For parallelogram ABCD below, find acute angle of <DBC to the nearest whole degree.
52. For the figure below, CD = 15, find BC to the nearest whole number.
53. A lighthouse is built on the edge of a cliff near the ocean, as shown in the accompanying diagram. From a boat located 200 feet from the base of the cliff, the angle of elevation to the top of the cliff is 18o and the angle of elevation to the top of the lighthouse is 28o. What is the height of the lighthouse, x, to the nearest tenth of a foot?