PROPOSITION XL. PROBLEM. 298. Upon a given side, homologous to a given side of a given polygon, to construct a polygon similar to the given polygon. E A B E B' Let ABCDE be the given polygon, and A'B' the given side. To construct upon the side A'B', homologous to AB, a polygon similar to ABCDE. Divide the polygon ABCDE into triangles by drawing the diagonals EB and EC. At A' construct ▲ B'A'E' = A, and draw B'E' making ▲ A'B'E' = 2 ABE. Then the triangle A'B'E' will be similar to ABE. (§ 256.) In like manner, construct the triangle B'C'E' similar to BCE, and the triangle C'D'E' similar to CDE. Then A'B'C'D'E' will be similar to ABCDE. For two polygons are similar when they are composed of the same number of triangles, similar each to each, and similarly placed (§ 266). EXERCISES. 75. To inscribe in a given circle a triangle similar to a given triangle. 76. To circumscribe about a given circle a triangle similar to a given triangle. NOTE. For additional exercises on Book III., see p. 226. BOOK IV. AREAS OF POLYGONS. PROPOSITION I. THEOREM. 299. Two rectangles having equal altitudes are to each other as their bases. NOTE. The word "rectangle," in the above statement, signifies the amount of surface of the rectangle. CASE I. When the bases are commensurable. Let ABCD and ABEF be two rectangles, having equal altitudes, and commensurable bases AD and AF. Let AG be a common measure of AD and AF, and let it be contained 7 times in AD, and 4 times in AF. Then, = AD 7 (1) Drawing perpendiculars to AD through the several points of division, the rectangle ABCD will be divided into 7 equal parts, of which ABEF will contain 4. (§ 113.) (2) Then, From (1) and (2), CASE II: When the bases are incommensurable. B G E A H F Let ABCD and ABEF be two rectangles, having equal altitudes, and incommensurable bases AD and AF. Let AD be divided into any number of equal parts, and let one of these parts be applied to AF as a measure. Since AD and AF are incommensurable, a certain number of the parts will extend from A to H, leaving a remainder, HF, less than one of the parts. Draw GH perpendicular to AD. Then, ABCD AD = ABGH AH (§ 299, Case I.) Now let the number of subdivisions of AD be indefinitely increased. Then the length of each part will be indefinitely diminished, and the remainder HF will approach the limit 0. By the Theorem of Limits, these limits are equal. (§ 188.) 300. COR. Since either side of a rectangle may be taken as the base, it follows that Two rectangles having equal bases are to each other as their altitudes. PROPOSITION II. THEOREM. 301. Any two rectangles are to each other as the products of their bases by their altitudes. Let A and B be any two rectangles, having the altitudes a and a', and the bases b and b', respectively. Let C be a rectangle having the altitude a and the base b'. Then the rectangles A and C, having equal altitudes, are to each other as their bases. ($ 299.) (1) And the rectangles C and B, having equal bases, are to. each other as their altitudes. ($ 300.) 302. SCH. It must be observed that the product of two lines signifies the product of their numerical measures when referred to a common unit. Ex. 1. A rectangular field is 182 feet long, and 102 feet wide. Another is 119 feet long, and 117 feet wide. their surfaces ? What is the ratio of DEFINITIONS. 303. The area of a surface is its ratio to another surface, called the unit of surface, adopted arbitrarily as the unit of measure (§ 179). Thus, if A represents a certain surface, and B the unit of The usual unit of surface is the square whose side is some linear unit; for example, a square inch, or a square foot. 304. Two surfaces are said to be equivalent when their areas are equal. The symbol is used for the words "is equivalent to." PROPOSITION III. THEOREM. 305. The area of a rectangle is equal to the product of its base and altitude. Let a and b be the altitude and base of the rectangle A; and let B be the unit of surface; i.e., a square whose side is the linear unit. Any two rectangles are to each other as the products of their bases by their altitudes (§ 301). |