36. A rhombus contains 100 square feet, and the shorter diagonal is 10 feet. Find the length of the other diagonal. 37. A rhombus and a square have equal perimeters. Which has the greater area? What is the ratio of their areas if the altitude of the rhombus is equal to one-half that of the square? 38. What ratio has the area of a rhombus whose acute angle is 60°, to that of a square having an equal perimeter? 39. A rhombus and a rectangle have equal bases and equal areas. Find their perimeters if one side of the rhombus is 5 feet, and the altitude of the rectangle is 3 feet. 40. The area of a rhombus is 60m, and the shorter diagonal is equal to one side of the rhombus. Find the perim eter. 41. The diagonals of a rhombus are 90 yards and 120 yards, respectively. Find the length of one side, and also the distance between the parallel sides. 42. Find the area of a parallelogram if the base is 40 feet 6 inches, and the altitude is 28 feet 9 inches. 43. If two parallelograms have equal areas, and the base of one is three times that of the other, what is the ratio of their altitudes? 44. What is the ratio of the areas of two parallelograms of given bases and altitudes? 45. What is the ratio of the areas of two triangles of given bases and altitudes? 46. If a, a' denote the bases, and h, h' the altitudes, respectively, of two equivalent triangles, state, in the form of a proportion, the relation of the four quantities a, a', h, h'. 47. A parallelogram and a triangle have equal altitudes, but the base of the parallelogram is equal to one-half that of the triangle. Compare their areas. Find the area of a triangle, given : 51. Rt. isos. A, hypotenuse 20 feet. 52. Rt. A, one leg 5 feet, hypotenuse 13 feet. 57. Equilat. A, altitude 24 feet. 58. The altitudes of two triangles are equal, and their bases are 12 feet and 16 feet, respectively. What is the ratio of their areas? 59. The altitudes of two triangles are equal, and their bases are 20 feet and 30 feet, respectively. What is the base of a triangle equivalent to their sum, and having an altitude one-fourth as great? 60. A house is 40 feet long, 30 feet wide, 25 feet high to the roof, and 35 feet high to the ridgepole. Find the number of square feet in the entire exterior surface. 61. What must be the length of the hypotenuse of a right triangle in order that its area may be 500gm ? 62. Find the area of a right triangle if the perimeter is 60 feet, and its sides are as 3:4: 5. 63. Find the area of a right triangle if its sides are as 3:45, and the altitude upon the hypotenuse is 12 feet. 64. The legs of a right triangle are 30 feet and 40 feet. Find the areas of the parts into which the triangle is divided by a perpendicular drawn from the vertex of the right angle to the hypotenuse. 65. Find the area of a right triangle if one leg is 15m, and the altitude upon the hypotenuse is 8m. 66. The area of a right triangle is 300m and the hypotenuse is equal to 50m. Find the legs. 67. The area of a triangle is 875 square feet. Find its base and its altitude if they are as 14: 5. 68. ABC is a triangle, and AD the perpendicular from A upon BC. If AD=13 feet, and the lengths of the perpendiculars from D to AB and AC are 5 feet and 10% feet, respectively, find the area of the triangle. 69. Find the area of a triangle if the three sides are 104 feet, 111 feet, and 175 feet, respectively. 70. How many square feet of carpet are required to cover a triangular floor whose sides measure, respectively, 26 feet, 35 feet, and 51 feet? 71. The two legs of a right triangle are 1 foot and 2 feet. Find the radius of the inscribed circle. 72. Given the sides a, b, c of a triangle; find the radii r, p of the circumscribed and inscribed circles. 73. The three sides of a triangle are: AB 100 feet, BC 89 feet, AC 21 feet. Find the length of the perpendicular from C to AB. 74. Find the area of a triangle if the perimeter is 14" and the radius of the inscribed circle is 1.07". Find the area of a trapezoid, given: 75. Bases 50 feet and 34 feet, altitude 25 feet. 76. Median 25 feet, altitude 12 feet. 77. The area of a trapezoid is 700 square feet, the bases are 30 feet and 40 feet, respectively. Find the distance between the bases. 78. A trapezoid contains 240 square feet, and its altitude is 16 feet. Find the two bases: (i.) if one is 3 feet longer than the other; (ii.) if they are in the ratio 2 : 3. 79. The value of a field in the shape of a trapezoid is $5800. The bases are 200 yards and 119 yards, and the distance between them is 110 yards. Find the value per acre. 80. The bases of a trapezoid are 32 feet and 20 feet. Each of the other sides is equal to 10 feet. Find the area of the trapezoid. 81. Find the area of a trapezoid if the altitude is equal to the median, the difference of the bases is 1 foot, and the greater base is equal to the hypotenuse of a right triangle whose legs are the smaller base and the altitude. 82. A lot of land has the shape of a trapezoid. Its bases are 100m and 40m. Each of the other sides is 50m. Find (i.) the area in ars of the trapezoid; (ii.) the area of the triangle formed by producing the equal sides to their intersection. 83. Find the area of a trapezoid considered as the sum of a triangle and a parallelogram. 84. Find the area of a trapezoid considered as the difference of two triangles. 85. A lot of land has the shape of a trapezium. One diagonal is 108 feet, and perpendiculars upon it from the opposite vertices are 55 feet 3 inches, and 60 feet 9 inches, respectively. What will the lot cost at 60 cents per square yard? 86. ABCD is a trapezium, having AB = 87 feet, BC 119 feet, CD = 41 feet, DA = 169 feet, AC= 200 feet. Find the area. = 87. Find the area of a tangent quadrilateral whose perimeter is equal to 400 feet, the radius of the circle being 25 feet. 88. Find the area of the polygon ABCDE (Fig. 90) if perpendiculars are dropped from the vertices to a line L in the plane of the figure, and the following lines are measured: 89. What is the side of a square equivalent to a rectangle 200 feet long and 32 feet wide? 90. The dimensions of a triangle are: base, 360 feet; altitude, 240 feet. Find the altitude of an equivalent parallelogram if its base is 270 feet. |