345. COR. Rectangles having equal bases are to each other as their altitudes. Ex. 1082. Ex. 1081. To divide a given rectangle into three equivalent parts. 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. R R' b Given rectangles R and R' having the bases b and b' and the altitudes a and a' respectively. Proof. Construct the rectangle Q, having the same base as R', and the same altitude as R. Ex. 1084. Find the ratio of a rectangle 4 by 5 ft., and a square having a side of 10 ft. 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. 347. The area of a rectangle is equal to the product of its base and altitude. R 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 the area. Ex. 1087. The area of a rectangle is 360 sq. in., and its base is 4 yd. Find the altitude. Ex. 1088. A rectangle has an area of 125 sq. in. and is five times as long as wide. Find the dimensions of the rec tangle. Ex. 1089. In the annexed diagram, all angles (4, B, C, etc.) are rt. angles. Find the area of the figure if (a) AB = 9, BC = 8, DE = 4, EF = 3. (b) AB= BC= a, FEED = b. 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. Proof. Draw AF LAB, 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. Q. E. D. 1. Parallelograms having equal bases and equal cs are equivalent. Any two parallelograms are to each other as odes of their bases and altitudes. Cos. 3. Parallelograms having equal bases are to each box as their altitudes. COR. 4. Parallelograms having equal altitudes are to osch other as their bases. x. 1090. Find the area of a parallelogram whose base is 15 in. and wse altitude is 2 ft. Ex. 1091. Two sides of a parallelogram are 15 and 20 respectively, and piclude an angle of 30°. Find the area. Ex. 1092. The sides of a parallelogram are 5 and 8 respectively, and the projection of 5 upon 8 is three. Find the area. Ex. 1093. To divide a parallelogram into three equivalent parts. Ex. 1094. The sides of a parallelogram are 13 and 14, and one diagonal equals 15. Find the area. Ex. 1095. The sides of a parallelogram are 4 in. and 5 in. and the angle included by them equal 45°. Find the area. PROPOSITION V. THEOREM 355. The area of a triangle is equal to one half the product of its base and altitude. A Given ▲ ABC, having base b and altitude h. Proof. Construct ABDC. The diagonal of a parallelogram divides it into two equal 356. COR. 1. Triangles having equal bases and equal altitudes are equivalent. 357. COR. 2. Any two triangles are to each other as the products of their bases and altitudes. 358. Cor. 3. Triangles having equal bases are to each other as their altitudes. 359. COR. 4. Triangles having equal altitudes are to each other as their bases. 360. COR. 5. If two triangles have a common base and their vertices lie in a line parallel to the base, the triangles are equivalent. 361. COR. 6. If A denotes the area, and 2s the perimeter of ▲ abc, then ▲ = √s(s — a) (s — b)(s — c) (336) 362. DEF. To transform a figure means to construct another figure equivalent to the given figure. Ex. 1096. What is the locus of the vertices of all the equivalent triangles constructed on the same base ? Ex. 1097. To transform a given triangle into a right triangle. Ex. 1098. To transform AABC into an isosceles triangle, having its base equal to AB. Ex. 1099. To transform ▲ ABC into an isosceles triangle, having an arm equal to AB. Ex. 1100. To transform a given triangle into another one, having given one side. |