PROPOSITION XIX 213. Theorem. Two similar polygons may be decomposed into the same number of triangles, similar each to each and similarly placed. PROPOSITION XX 214. Theorem. The perimeters of similar polygons are to each other as any two homologous sides. 174. The perimeters of two similar polygons are 76 and 114. If a diagonal of the first is 16, what is the homologous diagonal of the second? 175. A tree casts a shadow 100 feet long, and at the same time a vertical rod 6 feet high casts a shadow 8 ft. 4 in. long. What is the height of the tree? 176. In Prop. XIII, if DA: BF=3:1, what is the ratio of DB to BC? PROPOSITION XXI 215. Theorem. If in a right triangle a perpendicular is drawn from the vertex of the right angle to the hypotenuse, 1. The triangles formed are similar to the whole triangle and to each other. 2. The perpendicular is a mean proportional between the two segments of the hypotenuse. 3. Either side about the right angle is a mean proportional between the whole hypotenuse and the adjacent segment. PROPOSITION XXII 217. Theorem. If a tangent and a secant are drawn from a point to a circumference, the tangent is a mean proportional between the secant and its external seg PROPOSITION XXIII 220. Theorem. If two chords intersect within a circle, the product of the segments of one equals the product of the segments of the other. 221. Theorem. The product of two sides of a triangle equals the product of the diameter of the circumscribed circle and the perpendicular to the third side from the opposite vertex. 222. Cor. The diameter of a circle circumscribed about a triangle equals what? |