PROPOSITION V. THEOREM 406 The area of a triangle is equal to half the product of its base and altitude. A a b B HYPOTHESIS. ABC is a triangle, b its base, and a its altitude. 407 COROLLARY 1. Triangles having equal bases and equal altitudes are equivalent. 408 COROLLARY 2. Triangles having equal bases are to each other as their altitudes. 409 COROLLARY 3. Triangles having equal altitudes are to each other as their bases. 410 COROLLARY 4. Two triangles are to each other as the products of their bases and altitudes. EXERCISES 807 The area of a rhombus is equal to half the product of its diagonals. 808 If two triangles have two sides of the one equal to two sides of the other, each to each, and the included angles supplementary, they are equivalent. [§ 407.] 411 The area of a trapezoid is equal to the product of its altitude and half the sum of the bases. HYPOTHESIS. b and b' are the bases, and a is the altitude of the trapezoid ABCD. CONCLUSION. The area of ABCD ža (b+b′). PROOF Draw the diagonal BD. Then the area of the ▲ BDC = 1 a × b, a x b'. § 406 Ax. 1 Q. E. D. 412 COROLLARY. product of its altitude and median. § 211 The area of a trapezoid is equal to the 413 SCHOLIUM. The area of any polygon may be found by dividing the polygon into triangles. In practice, however, the following method is preferred. Draw the longest diagonal. To this diagonal draw perpendiculars from the remaining vertices. The polygon is thus decomposed into right triangles and trapezoids, whose bases and altitudes are measured. The areas of the right triangles and previous theorems. trapezoids are then computed by the PROPOSITION VII. THEOREM 414 Two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles. HYPOTHESIS. The A ABC and ADE have the A common. The AABC and ADC, having the same vertex C and their bases in the same line AB, have the same altitude. "A having equal altitudes are to each other as their bases.' Multiplying (1) and (2), member by member, § 409 § 337 Q. E. D. EXERCISE 809. If, in the above figure, AB = 10, AC = 12, AD = 6, and AE = 5, find the ratio of the triangle ABC to the triangle ADE. PROPOSITION VIII. THEOREM 415 Two similar triangles are to each other as the squares of any two homologous sides. 810 Two similar triangles are to each other as the squares of any two homologous altitudes. 811 The homologous bases of two similar triangles are as 2 : 1. Find the ratio of their areas. 812 Two homologous altitudes of similar triangles are 3 in. and 5 in. Find the ratio of their areas. 813 The areas of two similar triangles are 12 sq. ft. and 27 sq. ft.; the base of the first is 6 ft. Find the homologous base of the second. 814 The sides of a triangle are 3, 5, 6. Find the sides of a similar triangle whose area is 9 times as large. 416 Two similar polygons are to each other as the squares of any two homologous sides. HYPOTHESIS. ABCDE and A'B'C'D'E' are similar polygons whose areas are S and S'. CONCLUSION. S: S'AB2: A'B'2. PROOF By drawing the diagonals AC, AD, A'C', and A'D', the polygons are divided into the same number of similar triangles, similarly placed. § 363 "Two similar A are to each other as the squares of any two homologous 417 COROLLARY. Two similar polygons are to each other as the squares of any two homologous lines. |