Proposition 15. Problem. 390. To construct a rectangle equivalent to a given square, and having the sum of two adjacent sides equal to a given line. Cons. Upon AB as a diameter, describe a semicircle. Draw CD to AB, cutting the Oce at D, and draw DEL to AB. Then AE and EB are the base and altitude of the required rectangle. since DE is a mean proportional between AE and EB (325), 391. SCH. When the side of the square S exceeds half the line AB the problem is impossible. EXERCISES. 1. Construct a square equivalent to the sum of two squares whose sides are 6 and 8 inches. 2. Construct a square equivalent to the difference of two squares whose sides are 15 and 25 feet. 3. The perimeter of a rectangle is 144 feet, and the length is three times the altitude: find the area. 4. On a given straight line construct a triangle equal to a given triangle and having its vertex on a given straight line not parallel to the base. 5. Construct a parallelogram that shall be equal in area and perimeter to a given triangle. Proposition 16. Problem. 392. To construct a rectangle equivalent to a given square, and having the difference of two adjacent sides equal to a given line. Given, the square S, and the line AB the difference CD Cons. Upon AB as a diameter, describe a O. Through C and the centre of the the Oce at D and H. S draw CH, cutting Then CH and CD are the base and altitude of the required rectangle. a tangent is a mean proportional between the whole secant and the and the difference between CH and CD is DH = AB. ... CH X CD is the required rectangle. EXERCISES. Q. E. F. 1. The bases of a trapezoid are 14 and 16 feet; the nonparallel sides are each 6 feet: find the area of the trapezoid. 2. Construct a rhombus equal to a given parallelogram and having one of the sides of the parallelogram for one side of the rhombus. Proposition 17. Problem. 398. Two similar polygons being given, to construct a similar polygon equal to their sum. Given, two homologous sides P and Q of two similar polygons R and S. Required, to construct a similar polygon equivalent to their sum. Cons. Draw AB = P. At A construct the rt. ▲ A, draw A ACQ, and join BC. On BC, homologous to P and Q, construct a polygon T similar to R and S, as in (354). Then T is the polygon required. 394. SCH. To construct a polygon similar to two given similar polygons, and equivalent to their difference, we find the side of a square equivalent to the difference of the squares on P and Q (389), and on this side, homologous to P and Q, construct a polygon similar to the given polygons R and S (354). This will be the polygon required (379), Proposition 18. Problem. 395. To find two straight lines proportional to two given polygons. Given, two polygons R and S. Required, to find two st. lines proportional to R and S. Cons. Find two squares equiva lent to the given polygons R and S (386); let P and Q be the sides of these squares. Construct the rt. / A, draw AB P, and AC = Q. Join BC, and draw AD 1 to BC. Proof. Since AD is a nuse BC, from the rt. A on the hypote ... AB : AC = BD : DC, the sqs. of the sides about the rt. ▲ are proportional to the adj. segments and ... BD, DC are proportional to the areas of the given polygons. Q. E. F. EXERCISE. Bisect a quadrilateral by a straight line drawn from one of its vertices. Let ABCD be the quad.; bisect BD in E, let E lie between AC and B; through E draw EF to AC to meet BC in F; join CE, EA, AF, then AFCD = ABCD, Proposition 19. Problem. 396. To construct a square which shall have to a given square the ratio of two given lines. Given, the square S, and the On CB, or CB produced, take CH a side of S. Draw HE = to BA. Then CE is the side of the required square. S Proof. Since CD is a from the rt. /C to the hypote nuse AB, ... CA': CB' = AD: DB. (324 But CA: CB CE: CH, a st. line || to a side of a ▲ cuts the other two sides proportionally (298), 397. SCH. To construct a polygon similar to a given polygon S, and having to it the given ratio of P to Q, we find, as in (396), a side x so that x shall be to s (where s is a side of S) as P is to Q, and upon a as a side homologous to s, construct the R S X S polygon R similar to S (354); this will be the polygon required, (379) |