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Math 70 Exam 2 review Use only a compass and straightedge to complete the following constructions. 1. Duplicate the three line segments shown below. Label them as they are labeled in the figure. A B C D E F 2. Construct an angle congruent to P. P 3. Duplicate ABC below. B A C 4. Construct a perpendicular bisector that is congruent to segment r. r 5. Construct perpendicular bisectors to divide segment s into four congruent segments. s 6. Construct the median GA in FGH. F G H 7. Construct the midsegment AB in FGH where A is the midpoint of FG and B is the midpoint of GH. F G H 8. Construct a perpendicular from AB that passes through P. P A B 9. Construct the line perpendicular to line p at point Q. Q p 10. Name an altitude for BCD. B E D F G C 11. The diagram below shows an obtuse isosceles triangle FGH. a. Construct the altitude GI to side F H . b. How could the construction be checked to make sure it was an altitude? G F H 12. Use only a compass and straightedge to complete the following constructions. Construct the angle bisector of P. P 13. Is AD an angle bisector of BAC ? B 10 ft D 9 ft A C Use only a compass and straightedge to complete the following constructions. 14. Use perpendicular lines to construct a line that passes through point P and is parallel to line l. l P 15. Construct a rhombus with x as the length of each side and A as one of the acute angles. x A 16. Construct a kite using the segments with lengths x and y as sides. x y 17. Construct a rhombus with sides of length p. p 18. Does the given information determine a specific rhombus or could the constructed rhombus vary? If the constructed rhombus could vary, how will it vary? Using only a compass and straightedge, construct a rhombus with a 45-degree angle and sides of length DE. D E [A] The lengths of the sides of the rhombus will vary. [B] Both the angles and the lengths of the sides of the rhombus will vary. [C] The angles of the rhombus will vary. [D] A specific rhombus is determined. 19. Construct the incenter of DFG. F D G 20. Construct the circumcenter of JKL. K L J 21. If point P is the circumcenter of ABC and the length of segment AP is 11 inches, what is the length of segment CP ? 22. Is point P an incenter? Explain why or why not. B J K 11 P 10 10 A C L 23. Is point P a circumcenter? Explain why or why not. B 7 in. P 7 in. 7 in. A C 24. The circumcenter of a triangle is ___ [A] equidistant from only two of the vertices. [B] equidistant from only two of the sides. [C] equidistant from the sides. [D] equidistant from the vertices. 25. Construct a circle circumscribing the triangle. 26. Construct a circle inscribed in the triangle. 27. Julian wishes to center a butcher-block table in a kitchen’s work triangle, that is, at a location equidistant from the refrigerator, stove, and sink. Which point of concurrency does Julian need to locate? 28. The first-aid center of Mt. Thermopolis State Park needs to be at a point that is equidistant from three bike paths that intersect to form a triangle. At what point of concurrency should they locate the center so that in an emergency medical personnel will be able to get to any one of the paths by the shortest route possible? [A] circumcenter [B] orthocenter [C] incenter 29. Use only a compass and straightedge to complete the following constructions. Construct the centroid of JKL. L J K 30. The centroid of a triangle divides each median into two parts so that the distance from the centroid to the vertex is _____ the distance from the centroid to the midpoint of the opposite side. 31. Point P is the centroid of the triangle shown below. If BL is 27 cm, what is PL? B K J P C L A [A] 27 cm [B] 81 cm [C] 13.5 cm [D] 9 cm 32. Point P is the centroid of the triangle shown below. If BP is 22 inches, what is BL? B J K P A [A] 242 in. C L [B] 33 in. [C] 44 in. [D] 11 in. Find the missing measurements. 33. a = _____ 35. d = _____ e = _____ f = _____ 34. b = _____ 36. j = _____ Find the missing measurements. k = _____ l = _____ Tell whether it is possible to draw a triangle with the given side lengths. 37. 3 in., 4 in., 5 in. 38. 1 cm, 7 cm, 8 cm 39. 3 ft, 5 ft, 9 ft Arrange the letters in order from greatest value to least value. 40. 41. 42. 43. Find the value of x. N 84° x 121° O M 44. Refer to the figure and information given below. Give a congruence statement for two triangles in the figure and name the congruence conjecture that supports the congruence. H G J I GJ JI 45. AC DC and BC CE . Write a paragraph proof to show that ABC DEC. B D C A E 46. Determine what information you would need to know in order to use the SSS Congruence Conjecture to show that the triangles are congruent. B A C D [A] BAD CDB [B] AD BD [C] ADB CBD [D] AD CB 47. Determine which triangles in the figure of a triangle are congruent by SAA. A B C F D E [B] ADE EBA [C] ABF AFE [D] EFD AFE [A] ABF EDF 48. In the figure below, LJ bisects IJK and ILJ JLK. Find a congruence statement for the two triangles in the figure and name the congruence shortcut used. I L J K [A] ILJ KLJ; ASA [B] ILJ KLJ ; SAA [C] KLJ LIJ ; SAA [D] KLJ LIJ ; ASA 49. If GHI JKL, what six things can you conclude about the corresponding parts of the two triangles? 50. Given: BD bisects AC , AB BC Show: CBD ABD B A D C 51. Find the value of x so that ABC XYZ. b g b g mA 46 , BC 7 x 3 meters, mX 46 , AC 31 meters, YZ 6x 6 meters B [A] x 6 A Z C X [B] x 5 Y [C] x 3 [D] x 4 52. Use the given information to make a sketch of ABC and DEF. Mark the triangles with the given information. B E C F AB DE 53. Given: ABED AB ED YY Show: BC DC A D C B E 54. Which set of statements and reasons would correctly complete the flow chart proof? Given: AB ED AB ED Show: BC DC YY A D C B E AB ED Given AB || ED Given ABC EDC AIA 2) _________ BC DC CPCTC 1) __________ From the information given, determine which triangles, if any, are congruent. State the congruence conjecture that supports the congruence statement. If the triangles cannot be shown to be congruent from the information given, write “Cannot be determined.” 55. MXD ? Why? 56. BNG ? Why? 57.NMT = ? Why? 58.HOW ? Why? 59. Provide each missing reason or statement in the proof. Given: D C sDE sEC Show: sAE sBE Flow-chart Proof: GIVEN GIVEN 60. Provide each missing reason or statement in the proof. Given: Show: sACsBC, sAD sBE DCA BCE Flow-chart Proof: GIVEN BCE _______ GIVEN Math 70 Exam 2 review ANSWERS [1] A B C [6] D E F A F [2] G H [7] F [3] A B G H [4] [8] P r A B [5] s [9] A Q B p [15] [10] BG [11] a. G F I H [16] Sample answer: b. Using a protractor you could measure one of the angles at point I and show it is 90°. [12] P [17] [13] No [14] l P The angles of the rhombus will vary. [18] [D] [19] Sample answer: [26] F D G [27] circumcenter [20] Sample answer: K [28] [C] [29] L J [21] CP 11 in. [22] No. Point P is not equidistant from each side of [30] twice the triangle. [23] Yes. The distance from the circumcenter to the vertices is equidistant. [24] [D] [25] [31] [D] [32] [B] [33] a 80 [34] b 50 [35] d 30 , e 30 , f 14 cm [36] j 45 , k 45 , l 6 ft [37] Yes [38] No C F [39] No [40] a, b, c [41] c, b, a [42] c, b, a A B AB ED Given YY AB ED Given ABCEDC AIA Conj. ACBECD VA Conj. [45] ACB DCE because they are vertical angles. So ABC DEC by SAS. [54] 1) ACB ECD; VA Conj. 2) ACB ECD; SAA [46] [D] [47] [A] [55] ∆𝑀𝑌𝑇 , 𝑆𝐴𝑆 [48] [A] [49] GH J K , H I K L, GI J L, G J , H K, I L [50] We are given that BD bisects AC and AB BC. The definition of bisect tells us that AD CD, and BD BD because they are the same segment. The SSS shortcut gives ADB CDB, so CBD ABD by CPCTC. [51] [C] [52] Sample answer: E [53] [43] 155° [44] GHJ IHJ by the SAS Conjecture D [56] CANNOT BE DETERMINED [57] CANNOT BE DETERMINED [58] ∆𝑇𝑌𝑊 , 𝐴𝑆𝐴 [59] ASK [55] ASK ACBECD SAA