58. The areas of two are 144 square yards and 108 square yards. Two sides of the second are 12 yards and 21 yards, and one side of the first is 9 yards. Find a second side of the first, which, with the side 9 yards includes an equal to the of the second included by the sides 12 yards and 21 yards. 59. Find the area of a square whose diagonal is 8m. 60. Find the difference in perimeter between a rectangle whose base is 16 feet and an equivalent square whose side is 12 feet. 61. Find the diagonals of a rhombus whose side is 6 feet 1 inch and whose area is 9 square feet 24 square inches. 62. Find the area of a trapezoid whose parallel sides are 28 and 33", and whose non-parallel sides are 12" and 13m. 63. Find the dimensions of a rectangle whose area is 1,452 square feet and one of whose sides is its diagonal. 64. The sides of a are 26m, 28m, 30m; find its area, the three altitudes, and the radii of the inscribed, escribed, and circumscribed. 65. How many tiles, 6 inches by 4 inches, will it take to cover a swimming pool 40 feet by 27 feet? 66. Find the sides of an isosceles right whose area is 98a. 67. Find the area (in centares) and one side of a rhombus, if the sum of the diagonals is 34 feet and their ratio is 5: 12. 68. The bases of a trapezoid are 197.3" and 142.7m, and its area 37.57; find its altitude. (Log.) 69. Find the area, in square feet, of a right ▲, when the sides are in the ratio 3:4: 5, and the altitude to the hypotenuse is 1.2Dm. 70. In the quadrilateral A B C D, A B = 10", B C = 17", C D = 13", D A = 20m, and A C = 21m; find the area in hektares, and the perpendiculars from B and D to A C. 71. Find the area of a ▲ if the perimeter is 82 feet and the radius of the inscribed 1.3 feet. 72. Find the ratio of the areas of two equilateral A if the side of one is 10m and the altitude of the other is 10m. 73. Find the area, the altitudes, and the radii of the inscribed, escribed, and circumscribed O of the isosceles A whose leg is 5 feet 5 inches and whose base is 10 feet 6 inches. 74. The bases of a trapezoid are 13m and 61m; the nonparallel sides are 25 each; find the area of the trapezoid. 75. How many yards of carpet of a yard wide will it take to carpet a room 15 feet by 18 feet? 76. Find the area of a rhombus whose perimeter is 6TM and one of whose diagonals is 1.2m. 77. The altitude of a given is .32Km; find the homologous altitude, in miles, of a similar 49 times as large. 78. Find the area of a pentagon whose perimeter is 5.18, circumscribed about a O whose diameter is 1.1 ". 79. Find the area in square metres of a right ▲ in which a perpendicular from the vertex of the right to the hypotenuse divides the hypotenuse into segments of 3914 feet and 11 feet. (Log.) 80. Upon the diagonal of a rectangle 6m by 8m a ▲ whose area is three times the area of the rectangle is constructed; find the altitude of the A. 81. Find the side of an equilateral equivalent to the sum of two equilateral A whose sides are respectively 5TM and 12". 82. Find the area of a trapezoid whose bases are 26 feet and 40 feet, and whose other sides are 13 feet and 15 feet. 83. The three sides of a are 417.31 feet, 589.72 feet, and 389.6 feet; find its area in ares. (Log.) 84. Find the radii of the inscribed, escribed, and circumscribed. (Log.) 85. Find the three altitudes. (Log.) 86. Find the median to the longest side. 87. Find the bisectors of the three . (Log.) 88. The base of a ▲ is 25", its altitude 12m; find the area of the cut off by a line parallel to the base and two-thirds of the way from the vertex to the base. 89. Two homologous sides of two similar A are 12 feet and 35 feet, respectively; find the homologous side of a similar ▲ equivalent to their sum. altitude of the is 2m and the altitude of the are equal, and the 5m; find the ratio of their areas. 91. How many yards of wall paper are required to paper a room 25 feet long, 22 feet wide, and 12 feet high, allowing for a chimney which projects into the room 1 foot, one door 5 feet by 7 feet, another 10 feet by 10 feet, a mantel 4 feet by 6 feet, and a window 6 feet by 11 feet? 92. The homologous altitudes of two similar ▲ are 5TM and 15, respectively; what fraction of the second is the first? 93. Find the legs of a right whose hypotenuse is 25Hm and whose area is 150Ha. 94. In a whose base is 22 feet, find the length of the line parallel to the base and dividing the into two equal parts. (Log.) 95. Find the area of the whose sides are to each other as 5 12 13, and whose altitude to the greater side is 23 inches. 96. The area of the polygon P is 735.89m, and of the similar polygon Q is 98.474m; find the side of Q homologous to a side of P equal to 81.41m. (Log.) 97. If two sides of a whose area is 9 acres are 165 rods and 201 rods, what is the length of the portions of these sides cut off by a line parallel to the base and cutting off a of 4 acres? 98. Find the area of a right ▲ whose hypotenuse is 70m and one of whose is 60°. (Log.) 99. The side of a square is 12m; find the side of a square having the ratio 8 to 3 to this square. 100. In a trapezoid whose altitude is 10 feet and whose bases are 21 feet and 29 feet, what is the length of a line parallel to the bases and 2 feet from the smaller base. BOOK V. NOTE I. The answers to a large number of the problems of this Book may be left in an expressed form, if desired. For example: What is the area of a hexagon inscribed in a O whose radius is 15 feet? Ans. 6 × 152 √3. NOTE II. Quite a number of problems in this Book which seem difficult, on a mere reading, are rendered quite easy by drawing figures representing the given conditions and requirements. NOTE III.-In many of these problems it is well to represent the number in terms of which the answer is to be gotten by a letter, and then replace the letter by its value in the final form of the result, as in finding the area, etc., of circumscribed and inscribed polygons in terms of the radius. 1. How many degrees in each of a regular octagon ? Of a regular dodecagon? Of a regular polygon of 27 sides? 2. How many degrees in the at the centre of a regular polygon of 15 sides? Of 16 sides? 3. Find the side of a square inscribed in a O whose radius is 91 feet. 4. Find the radius of a O circumscribed about a regular hexagon whose perimeter is 5.1m. 5. How many degrees in each exterior of a regular polygon of 18 sides? Of 25 sides? Of 35 sides? 6. How many sides has the regular polygon whose at the centre is 17° 8' 7" ? |