Geometry GT Midterm Review PDF

Geometry GT Midterm Review Name: ___________________________ #1-13: Select the correct response for each of the following questions. 2 1. Which of the...

Geometry GT Midterm Review Name: ___________________________ #1-13: Select the correct response for each of the following questions. 2 1. Which of the following is the equation of a line perpendicular to the line 𝑦 = 4𝑥 + 3? a. 𝑦 = 4𝑥 − 3 c. 𝑦 = 2𝑥 − 2 3 1 b. 𝑦 = − 4 𝑥 − 3 2 d. 𝑦 = −2𝑥 − 3 2. Given ∠𝐴 ≅ ∠𝑍 and ∠𝐵 ≅ ∠𝑋, which of the following do you need to prove ∆𝐴𝐵𝐶 ≅ ∆𝑍𝑋𝑌 by ASA? a. ̅̅̅̅ 𝐴𝐵 ≅ ̅̅̅̅ 𝑍𝑋 c. ̅̅̅̅ 𝐴𝐶 ≅ ̅̅̅̅ 𝑍𝑌 ̅̅̅̅ ≅ 𝑋𝑌 b. 𝐴𝐶 ̅̅̅̅ ̅̅̅̅ ≅ 𝑋𝑌 d. 𝐴𝐵 ̅̅̅̅ 3. What is the best justification for ∆𝑀𝑇𝐻 ≅ ∆𝐺𝐻𝑇? a. ASA c. SAS b. HL d. SSA 4. If EFGH is a rectangle, what is FH? a. 4 c. √65 b. √33 d. 11 5. If 𝑊(−3, 1), 𝑋(2, 3), 𝑌(5, 0), what are the coordinates of the vertices of ∆𝑊𝑋𝑌 after a reflection over line 𝑥 = 2? a. W’ (3, -1), X’ (-2, -3). Y’ (-5, 0) c. W’ (3, 1), X’ (-2, 3). Y’ (-5, 0) b. W’ (7, 1), X’ (2, 3). Y’ (-1, 0) d. W’ (1, 7), X’ (3, 2). Y’ (0, -1) 6. M is the midpoint of 𝐴𝐷̅̅̅̅ and 𝐵𝐶 ̅̅̅̅. Which triangle congruence theorem could be used to prove that ∆𝐴𝐵𝑀 ≅ ∆𝐶𝐷𝑀? B D a. SSS c. ASA M b. SAS d. HL A C 7. Which translation of the segment with endpoints 𝐵(2, 8) 𝑎𝑛𝑑 𝐶(5,4) is shown by segment B’C’ in the diagram below? a. (𝑥, 𝑦) → (𝑥 − 4, 𝑦 − 1) 𝐵 𝐵′ b. (𝑥, 𝑦) → (𝑥 − 4, 𝑦 + 1) c. (𝑥, 𝑦) → (𝑥 + 4, 𝑦 − 1) 𝐶 d. (𝑥, 𝑦) → (𝑥 + 4, 𝑦 + 1) 𝐶′ 8. ∆𝐷′𝐸′𝐹′ is the image of ∆𝐷𝐸𝐹 under a 900 counterclockwise rotation with a center at the origin. If the triangle has the following coordinates D(1, 5), E(6,4), and F(3, 1) determine the coordinate of the vertices of ∆𝐷 ′ 𝐸 ′ 𝐹 ′ ? a. (-5, 1) (-4, 6) and (-1, 3) c. (5, 1) (4, 6) and (1, 3) b. (-5, -1) (-4, -6) and (-1, -3) d. (5, -1) (4, -6) and (1, -3)  9. Solve for x. a. 4 c. -8 b. -8, 8 d. -16, 4 𝑥 2 + 6𝑥 6𝑥 + 64 10. In this figure, which of the following best describes the relationship between ∠1 and ∠2? a. Adjacent and complementary. b. Adjacent and supplementary. c. Adjacent, complementary, and a linear pair. d. Adjacent, supplementary, and a linear pair. 11. In the diagram below, 𝑚∠𝐵𝐴𝐶 = (𝑥 2 − 2)0 and 𝑚∠𝐵𝐴𝐷 = (𝑥 2 + 7𝑥 − 10)0. Determine the 𝑚∠𝐵𝐴𝐷. a. 60 c. 340 b. 14 0 d. 680 12. In the diagram, 𝑝||𝑞, 𝑚∠1 = (3𝑥 2 + 20𝑥)0 𝑎𝑛𝑑 𝑚∠2 = (2𝑥 2 + 5𝑥)0. If 𝒙 is a 𝑝 positive integer, which of the following statements must be true about ∠3? 2 a. It is an acute angle. 1 b. It is an obtuse angle. 𝑞 c. It is a right angle. d. It is a straight angle. 3 *Picture is not drawn to scale 13. ∠𝐴𝐵𝐶 𝑎𝑛𝑑 ∠𝐶𝐵𝐷 are supplementary angles. If ∠𝐴𝐵𝐶 is an obtuse angle, which of the following statements about ∠𝐶𝐵𝐷 must be true? a. 𝑚∠𝐶𝐵𝐷 > 900 c. 𝑚∠𝐶𝐵𝐷 ≤ 900 b. 𝑚∠𝐶𝐵𝐷 = 𝑚∠𝐴𝐵𝐶 d. 𝑚∠𝐶𝐵𝐷 < 900 14. For parallelogram PQLM below, if 𝑚∠𝑃𝑀𝐿 = 830 , 𝑡ℎ𝑒𝑛 𝑚∠𝑃𝑄𝐿 = ________________. a. 830 c. 𝑚∠𝑃𝑄𝑀 b. 97 0 d. 𝑚∠𝑄𝐿𝑀 #15 -26: Answer each of the following multi-part questions. 15. Here are some transformation rules. For each rule, describe whether the transformation is a rigid motion, a dilation, or neither. a. (𝑥, 𝑦) → (2𝑥, 𝑦 + 2) b. (𝑥, 𝑦) → (2𝑥, 2𝑦) c. (𝑥, 𝑦) → (𝑥 + 2, 𝑦 + 2) d. (𝑥, 𝑦) → (𝑥 − 2, 𝑦) 16. Given that ∆𝐿𝑀𝑁 ≅ ∆𝑈𝑉𝑊, complete the statements: a. ̅̅̅̅̅ 𝑈𝑊 ≅ ______________ b. ∠𝐿𝑀𝑁 ≅ _______________ c. 𝐼𝑓 𝑁𝑀 = 13, then _____________ = 13 17. Write the equation, in slope-intercept form, of the line that is: a. Parallel to 2𝑥 + 2𝑦 = 12 and contains the point (−2, 7) b. Perpendicular to 𝑦 = 2𝑥 − 5 and contains the point (−1,4) 2 18. RECT is a rectangle and the slope of ̅̅̅̅ 𝑅𝐸 𝑖𝑠 5. ̅̅̅̅? How do you know? a. What is the slope of 𝑅𝑇 ̅̅̅̅ ? How do you know? b. What is the slope of 𝑇𝐶 19. Justify why each of the triangles below are congruent. Answer the questions. Write a congruence statement. a. Which angle corresponds to ∠𝐵𝐴𝐶? ̅. Which angle corresponds to d. K is the midpoint of 𝐽𝐿 ∠𝐽? b. ̅̅̅̅ 𝐸𝐺 bisects ∠𝐹𝐸𝐻 e. A is the midpoint of ̅̅̅̅ 𝑀𝑆, ∠𝑆 ≅ ∠𝑀 c. ̅̅̅̅ 𝑂𝑄 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ∠𝑃𝑄𝑁, ∠𝑃 ≅ ∠𝑁 f. ∠𝑅 𝑎𝑛𝑑 ∠𝑇 are right angles. What would prove ∠𝑅𝑈𝑆 ≅ ∠𝑇𝑈𝑆? 20. Solve for x: a. 8𝑥 − 4 + 3(𝑥 + 7) = 6𝑥 − 3(𝑥 − 3) b. 8𝑥 2 + 10𝑥 + 3 = 0 21. Simplify: a. √108 b. √192 22. Given the figure below and 𝑚∠𝐵𝐴𝐶 = (3𝑥 + 41)0 , 𝑚∠𝐴𝐵𝐶 = (12𝑥 − 31)0 , 𝑎𝑛𝑑 𝑚∠𝐵𝐶𝐴 = (7𝑥 − 6)0. a. Find x b. Classify ∆𝐴𝐵𝐶 by its sides and angles. c. Find 𝑚∠𝐵𝐶𝐷 23. In the given figure, R and S are on the perpendicular bisector of QT, and QR = 10, QT = 12. a. Find RT b. Find ST c. Find RS 24. Use the following diagram, where 𝑙||𝑚 to answer the following: a. What is the best justification for the conclusion that ∠1 ≅ ∠3? 𝑙 2 b. What is the best justification for the conclusion that ∠3 ≅ ∠7? 1 3 4 𝑚 c. What is the best justification for the conclusion that ∠4 ≅ ∠6? 6 5 7 d. What is the best justification for the conclusion that ∠2 ≅ ∠8? 8 e. If 𝑚∠3 = 3𝑥 2 + 20𝑥 + 94 and 𝑚∠6 = 𝑥 2 + 4𝑥 + 22, and x is a positive integer, classify ∠8. f. If 𝑚∠1 = 2𝑥 2 + 8𝑥 and 𝑚∠7 = 5𝑥 2 − 10𝑥, find 𝑚∠6. 25. Determine if a triangle can be created from the given 3 sides. a. √13, √65, 2√13 b. 3, √11, 7 c. 3√2, 6, 4√3 26. Using the figure below, where 𝑚∠1 = 610 , 𝑚∠𝐴𝐵𝐶 = 810 , 𝑎𝑛𝑑 𝑚∠3 = 420. a. Find 𝑚∠5 and 𝑚∠4. c. What is the shortest side of ∆𝐷𝐵𝐶? b. What is the longest side of ∆𝐴𝐵𝐷? d. Classify ∆𝐵𝐷𝐶 by sides and angles.  #27- 30: Write a two-column proof for each problem. 27. Given: ̅̅̅̅̅ 𝑁𝑀||𝑃𝑄̅̅̅̅ 29. Given: ∠𝑆 ≅ ∠𝑊, 𝑌 is the midpoint of ̅̅̅̅ 𝑇𝑉 Prove: ∆𝑁𝑀𝑂~∆𝑃𝑄𝑂 ̅̅̅̅ ≅ 𝑊𝑉 Prove: 𝑆𝑇 ̅̅̅̅̅ 28. Prove that ∆𝐴𝐵𝐶~∆𝐴𝐸𝐷 two different ways 30. Given: ̅̅̅̅ 𝐵𝐶 ≅ ̅̅̅̅ 𝐷𝐴, ̅̅̅̅ ̅̅̅̅ 𝐵𝐶 ||𝐷𝐴 Prove: E is the midpoint of ̅̅̅̅ 𝐶𝐴 #31-50: Solve each problem. 31. Graph the triangle whose vertices have coordinates: A(-8, 2), B(-2, 2) and C(-5, 7). Draw its reflection in the x-axis then the line 𝑥 = −6. 32. Given that ∠𝐵 ≅ ∠𝐸 𝑎𝑛𝑑 ∠𝐶 ≅ ∠𝐹, what other piece of information is needed to show that ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹 by the ASA ≅ Postulate? 33. Find the value of ̅𝐻𝐼 ̅̅̅ that makes ∆𝐻𝐼𝐽~∆𝐾𝐿𝑀. Round to the nearest tenth. 34. If ∆𝐴𝐵𝐶~∆𝐴𝐸𝐷, what is the length of ̅̅̅̅ 𝐵𝐶 ? Round to the nearest tenth. 35. ∆𝐽𝐺𝑋~∆𝑋𝑊𝑌. Find XY 36. Find x and y. 37. Farmer Gaston has a triangular field with two parallel irrigation pipes marked 𝑃1 and 𝑃2 on the map below. She wants to find the distance from barn B to the north end of pipe 2. The distance from the barn to the south end of pipe 1 is 100 feet. The distance from there to the south end of pipe 2 is 20 feet. The distance from the barn to the north end of pipe 1 is 114 feet. How far is the barn from the north end of pipe 2? Round your answer to the nearest foot. 38. What is the distance between (4, -7) and (-2, 10) to the nearest hundredth? 39. In the figure 𝑚∠𝐴𝐸𝐷 = 1050 , 𝑚∠𝐵𝐸𝐷 = 840 , 𝑎𝑛𝑑 𝑚∠𝐴𝐸𝐶 = 740. What is 𝑚∠𝐵𝐸𝐶? 40. If ⃗⃗⃗⃗⃗ 𝐴𝑃 bisects ∠𝑆𝐴𝑇. What is the 𝑚∠𝑆𝐴𝑃? *Picture is not drawn to scale 41. Solve for x. 42. Isosceles ∆𝐴𝐵𝐶 with vertex ∠𝐴 with 𝑚∠𝐴 = 250 and 𝑚∠𝐶 = (3𝑥 − 4)0. Find 𝑚∠𝐵𝐶𝐷. 43. In the diagram 𝑙||𝑚. Classify ∆𝐴𝐵𝐶. 44. If QT is a median of ∆𝑄𝑅𝑆, what is the perimeter of ∆𝑄𝑅𝑆? R l 5x-8 B 700 3x T 3x+6 A 1100 m Q C 5x S 45. Given: ABCD is a parallelogram 32 a. Find BE b. Find ED 46. Construct a line parallel to the given line through the given point.  47. Construct a line perpendicular to the given line through the given point. #48-50: Construct a regular hexagon, an equilateral triangle, and a square inscribed in a circle.