GEOMETRYMIDTERMREVIEW...
Transcript of GEOMETRYMIDTERMREVIEW...
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GEOMETRY MIDTERM REVIEW Name: _______________________________ Proving Triangles Congruent and Similar 1. A student would like to prove ΔACB ≅ ΔECD by SAS ≅ SAS . Using the two pairs of congruent sides, which of the following is the third piece of information needed? (1) AB ≅ DE (2) AB DE (3) AE and BD intersect at C (4) AE bisects BD at C 2. A student would like to prove ΔROS ≅ ΔQOP by SAS ≅ SAS . Using the pair of congruent sides and pair of congruent angles, which of the following is the third piece of information needed? (1) RS PQ
(2) RQ bisects PS (3) R ≅Q (4) PS bisects RQ 3. Which method proves ΔABC ≅ ΔEDC ? (1) HL ≅ HL (2) SAS ≅ SAS (3) AAS ≅ AAS (4) ASA ≅ ASA 4. Given BC ≅ DC , and AC bisects BCD , which method can be used to prove ΔBCA ≅ ΔDCA ? (1) HL ≅ HL (2) SAS ≅ SAS (3) AAS ≅ AAS (4) ASA ≅ ASA
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5. The diagram shows ΔNYQ ≅ ΔPYQ . Which statement is not true? (1) N ≅P (2) NYQ ≅PYQ
(3) YN ≅YP (4) YQ ≅QP 6. The diagram shows ΔQRS ≅ ΔXYZ . Which statement must be true? (1) QR ≅ XY (2) Q ≅Z (3) QRS ≅XZY (4) QS ≅ XY #7.
Given: MP and LO bisect each other Prove: LM ≅OP
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#8.
Given: BA ⊥ BC EF ⊥ ED AD ≅CF BAC ≅EFD Prove: BC ≅ ED
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Given: DE AC Prove: ΔBDE ~ ΔBAC
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Homework #1 – Proving Triangles Congruent and Similar 10. A student would like to prove ΔDAT ≅ ΔDET by SAS ≅ SAS . Using the two pairs of congruent sides, which of the following is the third piece of information needed? (1) DT ≅ DT (2) DA ⊥ AT , DE ⊥ ET (3) DT bisects ADE (4) DT bisects ATE 11. Which method proves ΔCAB ≅ ΔBDC ? (1) HL ≅ HL (2) SAS ≅ SAS (3) AAS ≅ AAS (4) ASA ≅ ASA 12. The diagram shows ΔCAF ≅ ΔBDE . Which statement must be true? (1) FAC ≅EBD (2) FCA ≅EBD (3) FC ≅ BD (4) AF ≅ BE #13.
Given: RS ⊥ ST , UT ⊥ ST ST bisects RU Prove: RS ≅UT
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#14.
Given: AC bisects BAD and BCD Prove: B ≅D
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Given: AB DE Prove: ΔABC ~ ΔEDC
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Transformations 15. If the point 3,−2( ) is rotated 180° about the origin, what are the coordinates of its image? 16. If the point −1, 3( ) is reflected through the y − axis , what are the coordinates of its image? 17. If the point −3, 7( ) is reflected through the line y = x , what are the coordinates of its image? 18. Which of the following transformations produces and image that is similar but not congruent? (1) T 3,−5 (2) R180 (3) D−2 (4) ry−axis 19. The image of square JKLM preserves which properties under the transformation T 1,3 ?
(1) parallelism only (3) both parallelism and orientation (2) orientation only (4) neither parallelism nor orientation 20. The image of rhombus MNOP preserves which properties under the transformation D2 ? (1) parallelism only (3) both parallelism and orientation (2) orientation only (4) neither parallelism nor orientation 21. In the diagram, ΔA 'B 'C ' is the image of ΔABC and ΔA ''B ''C '' is the image of ΔA 'B 'C ' . The composite transformation of ΔABC to ΔA ''B ''C '' is an example of a (1) reflection followed by a rotation (2) translation followed by a reflection (3) translation followed by a rotation (4) reflection followed by a translation
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22. In the diagram, A 'B ' is the image of AB and A ''B '' is the image of A 'B ' . The composite transformation of AB to A ''B '' is an example of a (1) translation followed by a rotation (2) rotation followed by a reflection (3) reflection followed by a rotation (4) translation followed by a reflection 23. Given the following illustration with the center of dilation at the origin, what is the scale factor that was used? (1) −2 (2) 2 (3) 3
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24. Given the following illustration with the center of dilation at the origin, what is the scale factor that was used? (1) −3 (2) 2 (3) 3 (4) −2
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25. Quadrilateral DEFG has vertices D 2, 3( ) , E 3,−1( ) , F −2,−3( ) , and G −3,2( ) . Graph and state the coordinates of D 'E 'F 'G ' , the image of DEFG after D3 . Is the resulting image an isometry? Explain.
26. ΔJKL has vertices J 1,2( ) , K 1,5( ) , and L 4,2( ) . Graph and state the coordinates of ΔJ ''K ''L '' , the image of ΔJKL after the transformation ry−axis rx−axis . State a single transformation that will map ΔJKL onto ΔJ ''K ''L '' .
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Homework #2 – Transformations 27. If the point −2,5( ) is rotated 90° about the origin, what are the coordinates of its image? 28. If the point −4,2( ) is rotated 270° about the origin, what are the coordinates of its image? 29. If the point 2,−6( ) is reflected through the x − axis , what are the coordinates of its image? 30. Under a transformation, ΔXYZ ≅ ΔX 'Y 'Z ' . This would be possible under each transformation except: (1) rx−axis (2) D1
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(3) R270 (4) T 1,3
31. The image of rectangle ABCD preserves which properties under the transformation R90 ? (1) parallelism only (3) both parallelism and orientation (2) orientation only (4) neither parallelism nor orientation 32. The image of ΔABC preserves which properties under the transformation rx−axis ? (1) parallelism only (3) both parallelism and orientation (2) orientation only (4) neither parallelism nor orientation 33. In the diagram, ΔX 'Y 'Z ' is the image of ΔXYZ and ΔX ''Y ''Z '' is the image of ΔX 'Y 'Z ' . The composite transformation of ΔXYZ to ΔX ''Y ''Z '' is an example of a (1) reflection followed by a rotation (2) translation followed by a reflection (3) reflection followed by a translation (4) translation followed by a reflection
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34. Given the following illustration with the center of dilation at the origin, what is the scale factor that was used? (1) −2 (2) 2 (3) 3
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35. Given the following illustration with the center of dilation at the origin, what is the scale factor that was used? (1) −2 (2) 2 (3) 3 (4) −3 36. ΔRST has vertices R −5,2( ) , S −4,5( ) , and T −1, 3( ) . Graph and state the coordinates of ΔR 'S 'T ' , the image of ΔRST after T 6,−4 . Is the resulting image an
isometry? Explain.
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37. ΔABC has vertices A −2,1( ) , B 0, 4( ) , and C 3,−1( ) . Graph and state the coordinates of ΔA ''B ''C '' , the image of ΔABC after the transformation R180 ry−axis . State a single transformation that will map ΔABC onto ΔA ''B ''C '' .
Constructions, Angles, and Similar Triangles 38. Using a compass and a straightedge, 39. Using a compass and a straightedge, construct the angle bisector of the construct the perpendicular bisector of angle shown below. line segment AB .
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40. Using a compass and a straightedge, construct an equilateral triangle with side MN . Using this triangle, construct a 30° angle with its vertex at M.
41. Using a compass and a straightedge, construct an equilateral triangle with side CD . Using this triangle, construct a 30° angle with its vertex at C.
42. Using the diagram below, if 1= 3x + 5 and 5 = 5x − 21 , for what value of x makes the lines parallel?
29. If the measures of three angles of a triangle are represented by x , 3x + 4 , and 4x , then the triangle must be (1) acute (3) isosceles (2) obtuse (4) right
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43. If two complementary angles are in the ratio of 2:3, how many degrees are in the measure of the larger angle? 44. In ΔABC , mA = 65° and mB = 85° . Which expression correctly represents the lengths of the sides of this triangle? (1) AB < BC < AC (3) BC < AB < AC (2) AB < AC < BC (4) AC < BC < AB 45. In ΔMNO , mM = 5x − 20 , mN = 2x + 8 , and mO = x . Which statement is true? (1) MN > NO (3) MN > MO (2) MN = MO (4) MN < NO 46. Solve for x and y.
47. In the diagram of ΔABC , D is the midpoint of AB , E is the midpoint of BC , and F is the midpoint of AC . If AB = 22 , BC =16 , and AC = 20 , what is the perimeter of parallelogram ADEF? 48. In the accompanying diagram, ΔABC is a right triangle. CD is the altitude to BA . If BD = 4 and DA = 9 . What is the length of CD ?
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49. In the accompanying diagram, ΔABC is a right triangle. CD is the altitude to BA . If BD = 2 and DA = 6 . What is the length of BC ? Homework #3 – Angles and Similar Triangles 50. Using the diagram below, if 2 = 2x +16 and 4 = 3x +14 , for what value of x makes the lines parallel?
51. If the measures of three angles of a triangle are represented by 2x , 4x − 6 , and 6x , then the triangle must be (1) acute (3) isosceles (2) obtuse (4) right 52. If two supplementary angles are in the ratio of 4:5, how many degrees are in the measure of the smaller angle? 53. In ΔDEF , mE =102° and mF = 46° . Which expression correctly represents the lengths of the sides of this triangle? (1) DE < EF < DF (3) EF < DE < DF (2) DE < DF < EF (4) DF < DE < EF 54. In ΔJKL , mJ = 3x +10 , mK = 2x , and mL = 5x − 30 . Which statement is not true? (1) JK = KL (3) mJ = mL (2) JK = JL (4) mJ < mK
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55. Solve for x and y.
56. In the diagram of ΔABC , D is the midpoint of AB , E is the midpoint of BC , and F is the midpoint of AC . If AB =16 , BC =10 , and AC =14 , what is the perimeter of ΔDEF ?
57. In the diagram of ΔABC , D is the midpoint of AB , E is the midpoint of BC , and F is the midpoint of AC . If AB = 40 , BC = 28 , and AC = 34 , what is the perimeter of parallelogram DECF?
58. In the accompanying diagram, ΔABC is a right triangle. CD is the altitude to BA . If BC = 6 and BA = 9 . What is the length of BD ?
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Right Triangles and Trigonometry 59. The length and width of a rectangle are 30 centimeters and 72 centimeters. What is the length of its diagonal? 60. According to the diagram below, what is the height of the plane flying in the air?
61. According to the diagram below, what is the distance between the beach and the top of the building?
62. Which equation could be used to find the measure of angle A in the right triangle show below?
(1) sinA = 86 (3) sinA = 10
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(2) cosA = 610
(4) tanA = 810
63. Which equation could be used to find the measure of angle C in the right triangle shown below?
(1) sinC =810
(3) cosC =106
(2) tanC =68 (4) tanC =
86
x
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200 ft.
64. A tree stands 20 meters tall and is located 30 meters from point A on the ground. Find the measure of the angle of elevation, θ .
65. A man flies a kite with a 100 foot string. The angle of elevation of the string is 52° . How high off the ground is the kite? 66. As shown in the diagram below, 40 foot ladder is placed against the side of a building, 25 feet from the base of the building. To the nearest tenth of a foot, find the height, h, the ladder reaches on the side of the building. To the nearest degree, find the angle, x, the ladder makes with the ground.
67. An airplane takes of 200 feet in front of a 60 foot building, as shown in the diagram below. To the nearest tenth of a foot, find the distance between the plane and the roof of the building. To the nearest degree, find the angle of elevation, x.
x