BOOK IV AREAS OF POLYGONS 341. DEF. The unit of surface is a square whose side is the unit of length, etc. Thus, a square 1 in. long and 1 in. wide is a square inch. Or a square 1 yd. long and 1 yd. wide is a square yard. 342. The area of a surface is the number of units of surface it contains. Thus, if the floor of a room is 25 ft. long and 15 ft. wide, it contains 15 x 25 or 375 sq. ft. Hence the area of the floor is 375 sq. ft. 343. Two figures are equivalent or equal if their areas are equal. MNOP Thus, if the area of ABC = 25 sq. ft., and the area of = 25 sq. ft., then ∆ABC is equivalent to ☐ MNOP, or in symbols: NOTE. The symbol = refers to the size of figures, while the symbol ~ relates to their shape. The symbol of congruence (=) relates to both size and shape. The symbol ~ has no meaning for figures that cannot differ in shape, e.g. for straight lines. Thence the symbol of congruence () when applied to straight lines has the same significance as the symbol of equality (=). If the symbol of equality (=) refers to areas, it may read either "is equivalent to" or "equals." Since many authors, however, designate congruent figures as equal figures, confusion may be avoided by giving preference to the term equivalent. The use of a particular symbol for equivalence () cannot be recommended. PROPOSITION I. THEOREM 344. Rectangles having equal altitudes are to each other as their bases. Given Rand R' the areas of two rectangles whose common altitude equals h, and whose bases are respectively b and b'. b' is divided into n equal parts, and one of these parts is laid off on b, b contains m such parts. If perpendiculars are erected at the points of division, Ris divided into m rectangles and R' is divided into n rectangles. b CASE II. is an irrational number. Since any approxi mate value of is a rational number, it must equal the corre b' R sponding approximate value of 2. (Case I.) Hence all R 345. COR. Rectangles having equal bases are to each other as their altitudes. Ex. 1081. To divide a given rectangle into three equivalent parts. Ex. 1082. From a given rectangle, to cut off another rectangle whose area is of the given one. Ex. 1083. To construct a rectangle, which is to a given one as m:n, when m and n are two given lines. PROPOSITION II. THEOREM 346. The areas of two rectangles are to each other as the products of their bases and altitudes. Given rectangles R and R' having the bases b and b' and the Proof. Construct the rectangle Q, having the same base as Ex. 1084. Find the ratio of a rectangle 4 by 5 ft., and a square having Ex. 1085. The diagonal of a rectangle is 26 in. long, and one of its sides 24 in. The diagonal of another rectangle is 25 in. long, and one of its sides 20 in. Find the ratio of the areas of the two rectangles. a side of 10 ft. PROPOSITION III. THEOREM 347. The area of a rectangle is equal to the product of its base and altitude. Given R a rectangle with base b and altitude a. 348. COR. The area of a square is equal to the square of its side. 349. REMARK. If the base and altitude of a rectangle are expressed by integral numbers, Prop. III may be proved by dividing the figure into squares. Thus, if the base contains 5 and the altitude 3 linear units, the figure can be divided into fifteen squares, each being the unit of surface. Ex. 1086. A rectangular field is 24 yd. long and 15 yd. wide. Find Ex. 1087. The area of a rectangle is 360 sq. in., and its base is 4 yd. Find the altitude. the area. Ex. 1088. A rectangle has an area of 125 sq. in. and is five times as long as wide. Find the dimensions of the rectangle. A F Ex. 1089. In the annexed diagram, all angles (A, B, C, etc.) are rt. angles. Find the area of the figure if (a) AB = 9, BC = 8, DE = 4, EF = 3. B C PROPOSITION IV. THEOREM 350. The area of a parallelogram is equal to the product of its base and its altitude. Given parallelogram ABCD having the base AB = b, and the altitude BE =h. To prove ABCD = bxh. Proof. Draw AF ⊥ AB, meeting CD produced in F. Then ABEF is a rectangle, having its base b and its alti * Obviously figures that are congruent must also be equivalent. (347) Q. E. D. 1 |