Ex. 293. The square constructed upon the sum of two straight lines is equivalent to the sum of the squares constructed upon these two lines, increased by twice the rectangle of these lines. A B C Let AB and BC be the two straight lines, and AC their sum. Construct the squares ACGK and ABED upon AC and AB respectively. Prolong BE and DE until they meet KG and CG respectively. Then we have the square EFGH, with sides each equal to BC. Hence, the square ACGK is the sum of the squares ABED D and EFGH, and the rectangles DEHK and BCFE, the dimensions of which are equal to AB and BC. F E K H G Ex. 294. The square constructed upon the difference of two straight lines is equivalent to the sum of the squares constructed upon these two lines, diminished by twice the rectangle of these lines. H K Let AB and AC' be the two straight lines, and BC their difference. Construct the square ABFG upon AB, the square ACKH upon AC, and the square BEDC upon BC (as shown in the figure). Prolong ED until it meets AG in L. C A B L E D G F The dimensions of the rectangles LEFG and HKDL are AB and AC, and the square BCDE is evidently the difference between the whole figure and the sum of these rectangles; that is, the square constructed upon BC is equivalent to the sum of the squares constructed upon AB and AC diminished by twice the rectangle of AB and AC. Ex. 295. The difference between the squares constructed upon two straight lines is equivalent to the rectangle of the sum and difference of these lines. I K H E D Let ABDE and BCGF be the squares constructed upon the two straight lines AB and BC. The difference between these squares is the polygon ACGFDE, which polygon, by prolonging CG to H, is seen to be composed of the rectangles ACHE and GFDH. Prolong AE and CH to I and K respectively, making EI and HK each equal to BC, and draw IK. The rectangles GFDH and EHKI are equal. The difference between the squares ABDE and BCGF is then equivalent to the rectangle ACKI, which has for dimensions AI AB + BC, and EH PROBLEMS OF CONSTRUCTION. PROPOSITION XI. PROBLEM. 381. To construct a square equivalent to the sum of two given squares. B S R' R Let R and R' be two given squares. Construction. Construct the rt. Z A. 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 to BC. Proof. S is the square required. BC AC+AB, § 379 (the square on the hypotenuse of a rt. ▲ is equivalent to the sum of the squares on the two sides). .. S≈ R' + R. Q. E. F. Ex. 296. If the perimeter of a rectangle is 72 feet, and the length is equal to twice the width, find the area. Ex. 297. How many tiles 9 inches long and 4 inches wide will be required to pave a path 8 feet wide surrounding a rectangular court 120 feet long and 36 feet wide? Ex. 298. The is equal to 5 feet. bases of a trapezoid are 16 feet and 10 feet; each leg Find the area of the trapezoid. PROPOSITION XII. PROBLEM. 382. To construct a square equivalent to the difference of two given squares. Let R be the smaller square and R' the larger. 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. Sis the square required. Proof. AC2 ≈ BC2 – 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 2 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 diagonal is 10 feet. Find the length of the other diagonal. PROPOSITION XIII. PROBLEM. 383. To construct a square equivalent to the sum of any number of given squares. Let m, n, o, p, r be sides of the given squares. To construct a square ≈≈ m2 + n2 + o2 + p2 + r2. Take AB = m. Draw AC =n and to AB at A, and draw BC. Draw CE =0 and to BC at C, and draw BE. Draw EF =p and 1 to BE at E, and draw BF. Draw FH=r and 1 to BF at F, and draw BH. The square constructed on BH is the square required. Proof. BH'FH2 + BF2, ≈ FH2 + EF2 + EB2, ≈ FH2 + EF2 + EC2 + CB2, ≈ FH2 + EC2 + EF2 + CA2 + AB, § 379 (the sum of the squares on the two legs of a rt. A is equivalent to the square on the hypotenuse). That is, BH2 ≈ m2 + n2 + o2 + p2 + 22. Q. E. F. PROPOSITION XIV. PROBLEM. 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 PH= A'B', and_PO= AB. Draw OH, and take A"B": - ОН. Upon A"B", homologous to AB, construct R" similar to R. Then R" is the polygon required. Proof. R' : R = A'B' : AB2, § 376 (similar polygons are to each other as the squares of their homologous sides). |