PROPOSITION XVII 209. Theorem. The homologous altitudes of similar triangles are in the same ratio as any two homologous sides. Appl. Dem. AA AD and 1 4 are homol. altitudes. 3 171. In a triangle ABC, AB = 40, BC = 50, and CA = = 70. Find the external segments of CA, formed by a bisector of the exterior angle at B. 172. In the diagram of Prop. XI, if DB: DA 4:5 and AC = 36, what is the length of EC? 173. In Prop. XIII, if AB, AC, and DB are 7, 9, and 12, respectively, what is BC? PROPOSITION XVIII 210. Theorem. Two polygons are similar if they are composed of the same number of triangles similar each to each and similarly placed. 211. Cor. I. Homologous diagonals in similar polygons have the same ratio as any two homologous sides. 212. Cor. II. Homologous lines, of any kind, in similar polygons have the ratio of similitude of the polygons. 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. If a 174. The perimeters of two similar polygons are 76 and 114. 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. |