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CHAPTER 9 Areas
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9.30. Provide the proofs requested in Fig. 9-23.
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(9.9)
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Fig. 9-23
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9.31. Provide the proofs requested in Fig. 9-24.
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(9.9)
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Fig. 9-24
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9.32. Prove each of the following: (a) A median divides a triangle into two triangles having equal areas.
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(9.10)
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(b) Triangles are equal in area if they have a common base and their vertices lie in a line parallel to the base. (c) In a triangle, if lines are drawn from a vertex to the trisection points of the opposite sides, the area of the triangle is trisected. (d) In trapezoid ABCD, base AD is twice base BC. If M is the midpoint of AD, then ABCM and BCDM are parallelograms which are equal in area. 9.33. (a) In ^ABC, E is a point on BM, the median to AC. Prove that area(^BEA) area(^BEC).
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(b) In ^ABC, Q is a point on BC, M is the midpoint of AB, and P is the midpoint of AC. Prove that area(^BQM) area(^PQC) area(quadrilateral APQM). area (^ACD). (c) In quadrilateral ABCD, diagonal AC bisects diagonal BD. Prove that area(^ABC)
(d) Prove that the diagonals of a parallelogram divide the parallelogram into four triangles which are equal in area. (9.10) 9.34. Find the ratio of the areas of two similar triangles if the ratio of two corresponding sides is (a) 1:7; (b) 7:2; (c) 1: 23; (d) a:5a; (e) 9: x; (f) 3: 2x; (g) s:s 22. (9.11) 9.35. Find the ratio of the areas of two similar triangles (a) If the ratio of the lengths of two corresponding medians is 7:10 (b) If the length of an altitude of the first is two-thirds of a corresponding altitude of the second (c) If two corresponding angle bisectors have lengths 10 and 12 (d) If the length of each side of the first is one-third the length of each corresponding side of the second (e) If the radii of their circumscribed circles are 71 and 5 2 (f) If their perimeters are 30 and 30 22 (9.11)
CHAPTER 9 Areas
9.36. Find the ratio of any two corresponding sides of two similar triangles if the ratio of their areas is (a) 100:1; (b) 1:49; (c) 400:81; (d) 25:121; (e) 4:y2; (f) 9x2:1; (g) 3:4; (h) 1:2; (i) x2:5; ( j ) x:16. (9.11) 9.37. In two similar triangles, find the ratio of the lengths of (a) Corresponding sides if the areas are 72 and 50 (b) Corresponding medians if the ratio of the areas is 9:49 (c) Corresponding altitudes if the areas are 18 and 6 (d) The perimeters if the areas are 50 and 40 (e) Radii of the inscribed circles if the ratio of the areas is 1:3 9.38. The areas of two similar triangles are in the ratio of 25:16. Find (a) The length of a side of the larger if the corresponding side of the smaller has length 80 (b) The length of a median of the larger if the corresponding median of the smaller has length 10 (c) The length of an angle bisector of the smaller if the corresponding angle bisector of the larger has length 15 (d) The perimeter of the smaller if the perimeter of the larger is 125 (e) The circumference of the inscribed circle of the larger if the circumference of the inscribed circle of the smaller is 84 ( f ) The diameter of the circumscribed circle of the smaller if the diameter of the circumscribed circle of the larger is 22.5 (g) The length of an altitude of the larger if the corresponding altitude of the smaller has length 16 23 9.39. (a) The areas of two similar triangles are 36 and 25. If a median of the smaller triangle has length 10, find the length of the corresponding median of the larger. (9.12) (b) Corresponding altitudes of two similar triangles have lengths 3 and 4. If the area of the larger triangle is 112, find the area of the smaller. (c) Two similar polygons have perimeters of 32 and 24. If the area of the smaller is 27, find the area of the larger. (d) The areas of two similar pentagons are 88 and 22. If a diagonal of the larger has length 5, find the length of the corresponding diagonal of the smaller. (e) In two similar polygons, the ratio of the lengths of two corresponding sides is 23:1. If the area of the smaller is 15, find the area of the larger. (9.11) (9.11)
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