PROPOSITION XI. PROBLEM. 381. To construct a square equivalent to the of two given squares. Take AC equal to a side of R', AB equal to a side of R; and draw BC. Construct the square S, having each of its sides equal S is the square required. (the square on the hypotenuse of a rt. ▲ is equivalent to the su squares on the two sides). ..SR'+R. Ex. 296. If the perimeter of a rectangle is 72 feet, and the 1 equal to twice the width, find the area. Ex. 297. How many tiles 9 inches long and 4 inches wide required to pave a path 8 feet wide surrounding a rectangular c feet long and 36 feet wide? Ex. 298. The bases of a trapezoid are 16 feet and 10 feet; 382. To construct a square equivalent to the difference of two given squares. Let R be the smaller square and R' the larger. To construct a square equivalent to R' — R. Construction. Construct the rt. Z A. Take AB equal to a side of R. From B as a centre, with a radius equal to a side of R', describe an arc cutting the line AX at C. Construct the square S, having each of its sides equal to AC. S is the square required. Proof. 2 AC BC-AB3, $380 (the square on either leg of a rt. ▲ is equivalent to the difference of the squares on the hypotenuse and the other leg). .. SR' R. Q. E. F. Ex. 299. Construct a square equivalent to the sum of two squares whose sides are 3 inches and 4 inches. Ex. 300. Construct a square equivalent to the difference of two squares whose sides are 24 inches and 2 inches. Ex. 301. Find the side of a square equivalent to the sum of two squares whose sides are 24 feet and 32 feet. Ex. 302. Find the side of a square equivalent to the difference of two squares whose sides are 24 feet and 40 feet. Ex. 303. A rhombus contains 100 square feet, and the length of one dieconel is 10 foot Find the length of the other diegenel 383. To construct a square equivalent to the of any number of given squares. Let m, n, o, p, r be sides of the given squares. To construct a square Construction. The m2 + n2 + o2 + p2 + r2. Take AB = m. Draw AC =n and I to AB at A, and draw B Draw CE = 0 and 1 to BC at C, and draw E BE at E, and draw E Draw FH- r and 1 to BF at F, and draw E square constructed on BH is the square requir Proof. BH'FH2 + BF2, ≈ FH2 + EF2 + EB2, ≈ FH2+ EF2+ EC2 + CB3, ≈ FH2 + EC2 + EF2 + CA2 + AB2, (the sum of the squares on the two legs of a rt. A is equivalent to t That is, on the hypotenuse). BH2 m2 + n2 + o2 + p2 + r22, 384. To construct a polygon similar to two given similar polygons and equivalent to their sum. Let R and R' be two similar polygons, and AB and A'B' two homologous sides. To construct a similar polygon equivalent to R + R'. Construction. Construct the rt. Z P. Take PHA'B', and PO= AB. Draw OH, and take A"B" = OH. Upon A"B", homologous to AB, construct R" similar to R. Then R is the polygon required. Proof. PO+PH' = OH', :. AB2 + A'B'2 = A"B"2. (similar polygons are to each other as the squares of their homologous sides). 385. To construct a polygon similar to two similar polygons and equivalent to their diffe Let R and R' be two similar polygons, and A'B' two homologous sides. To construct a similar polygon equivalent to R'R Construction. Construct the rt. Z P, and take PO = AB. From O as a centre, with a radius equal to A'l describe an arc cutting PX at H, and join O Take A"B" = PH, and on A"B", homologous to construct R" similar to R. Then R" is the polygon required. Proof. PH OH - OP', .. A"B"2 = A'B'2 —, (similar polygons are to each other as the squares of their homolog |