ABGH is a unit of measure of the given rectangles Let it be contained a certain no. of times to the point H with a remainder HF, < measure PROPOSITION III 242. Theorem. Any two rectangles are to each other as the products of their bases by their altitudes. 243. Theorem. having having = = multiply] bases alt. PROPOSITION IV The area of a rectangle equals the product of its base and altitude. [The area of the rectangle is to the square unit, taken as a standard, as the product of its base and altitude is to unity.] 244. Cor. The area of a square is the square of one of its sides. 245. Sch. When the sides of the rectangle are multiples of the linear unit, the truth of the theorem is illus trated and made evident by the accompanying figure. PROPOSITION V 246. Theorem. The area of a parallelogram equals the product of its base and altitude. 247. Cor. I. Parallelograms having equal bases and equal altitudes are equivalent. 248. Cor. II. Parallelograms having equal bases are to each other as their altitudes and those having equal altitudes are to each other as their bases. 249. Cor. III. Any parallelograms are to each other as the products of their bases and altitudes. EXERCISES 230. If the base of a rectangle is 2 ft. 8 in., and its altitude 1 ft. 6 in., what is the side of an equivalent square? 231. The area of a rectangle is 2520 sq. in., and its altitude is 11 yd. What is its perimeter in feet? 250. Theorem. PROPOSITION VI The area of a triangle equals one-half the product of its base and altitude. 251. Cor. I. Triangles having equal bases are to each other as their altitudes, and those having equal altitudes are as their bases. 252. Cor. II. Triangles are to each other as the products of their bases and altitudes (the factor disappears in the proportion). EXERCISES 232. The base of a rectangle is 4 times its altitude, and its area is 21 sq. ft. 112 sq. in. What are its base and altitude? 233. If the altitude of a trapezoid is 20 in. and the bases 2 ft. 4 in. and 6 ft., respectively, what is its area? 234. If the area of a trapezoid is 3136 sq. in., the altitude 2 ft. 8 in., and one base 6 ft. 8 in., what is the other base? 235. The diagonals of a parallelogram divide it into four equivalent triangles. 236. Lines drawn from two opposite vertices of a parallelogram to any point in the diagonal joining the other vertices, divide the parallelogram into two pairs of equivalent triangles. PROPOSITION VII 253. Theorem. The area of a trapezoid equals the product of its altitude and the half-sum of its bases. 254. Cor. The area of a trapezoid is equal to the product of its altitude and its median. [median equals one half the sum of the bases] EXERCISES 237. The area of a rhombus is equal to one-half the product of its diagonals. 238. If the segments of the hypotenuse of a right triangle formed by a perpendicular from the vertex of the right angle are x and y, what is the area of the triangle? What is the area of each of the triangles formed by the perpendicular? |