NUMERICAL EXERCISES. 324. Find the area of a rhombus, if the sum of its diagonals is 12 feet, and their ratio is 3; 5. 325. Find the area of an isosceles right triangle if the hypotenuse is 20 feet. 326. In a right triangle, the hypotenuse is 13 feet, one leg is 5 feet. Find the area. = h. 327. Find the area of an isosceles triangle if the base = 6, and leg = c. 328. Find the area of an equilateral triangle if one side = d. 329. Find the area of an equilateral triangle if the altitude 330. A house is 40 feet long, 30 feet wide, 25 feet high to the roof, and 35 feet high to the ridge-pole. Find the number of square feet in its entire exterior surface. 331. The sides of a right triangle are as 3:4:5. The altitude upon the hypotenuse is 12 feet. Find the area. 332. Find the area of a right triangle if one leg = a, and the altitude upon the hypotenuse = h. 333. Find the area of a triangle if the lengths of the sides are 104 feet, 111 feet, and 175 feet. 334. The area of a trapezoid is 700 square feet. The bases are 30 feet and 40 feet respectively. Find the distance between the bases. 335. ABCD is a trapezium; AB=87 feet, BC=119 feet, CD = 41 feet, DA 169 feet, AC 200 feet. Find the area. = 336. What is the area of a quadrilateral circumscribed about a circle whose radius is 25 feet, if the perimeter of the quadrilateral is 400 feet? What is the area of a hexagon having an equal perimeter and circumscribed about the same circle? 337. The base of a triangle is 15 feet, and its altitude is 8 feet. Find the perimeter of an equivalent rhombus if the altitude is 6 feet. 338. Upon the diagonal of a rectangle 24 feet by 10 feet a triangle equivalent to the rectangle is constructed. What is its altitude? 339. Find the side of a square equivalent to a trapezoid whose bases are 56 feet and 44 feet, and each leg is 10 feet. 340. Through a point P in the side AB of a triangle ABC, a line is drawn parallel to BC, and so as to divide the triangle into two equivalent parts. Find the value of AP in terms of AB. 341. What part of a parallelogram is the triangle cut off by a line drawn from one vertex to the middle point of one of the opposite sides? 342. In two similar polygons, two homologous sides are 15 feet and 25 feet. The area of the first polygon is 450 square feet. Find the area of the other polygon. 343. The base of a triangle is 32 feet, its altitude 20 feet. What is the area of the triangle cut off by drawing a line parallel to the base and at a distance of 15 feet from the base? 344. The sides of two equilateral triangles are 3 feet and 4 feet. Find the side of an equilateral triangle equivalent to their sum. 345. If the side of one equilateral triangle is equal to the altitude of another, what is the ratio of their areas? 346. The sides of a triangle are 10 feet, 17 feet, and 21 feet. Find the areas of the parts into which the triangle is divided by bisecting the angle formed by the first two sides. 347. In a trapezoid, one base is 10 feet, the altitude is 4 feet, the area is 32 square feet. Find the length of a line drawn between the legs parallel to the base and distant 1 foot from it. 348. If the altitude h of a triangle is increased by a length m, how much must be taken from the base a in order that the area may remain the same? 349. Find the area of a right triangle, having given the segments p, q, into which the hypotenuse is divided by a perpendicular drawn to the hypotenuse from the vertex of the right angle. PROBLEMS. 350. To construct a triangle equivalent to a given triangle, and having one side equal to a given length 7. 351. To transform a triangle into an equivalent right triangle. 352. To transform a triangle into an equivalent isosceles triangle. 353. To transform a triangle ABC into an equivalent triangle, having one side equal to a given length 7, and one angle equal to angle BAC. HINTS. Upon AB (produced if necessary), take AD = 1, draw BE to CD, and meeting AC (produced if necessary) at E; ▲ BED≈▲ BEC. 354. To transform a given triangle into an equivalent right triangle, having one leg equal to a given length. 355. To transform a given triangle into an equivalent right triangle, having the hypotenuse equal to a given length. 356. To transform a given triangle into an equivalent isosceles triangle, having the base equal to a given length. To construct a triangle equivalent to: 357. The sum of two given triangles. 358. The difference of two given triangles. 359. To transform a given triangle into an equivalent equilateral triangle. To transform a parallelogram into : 360. A parallelogram having one side equal to a given length. 361. A parallelogram having one angle equal to a given angle. 362. A rectangle having a given altitude. To transform a square into: 363. An equilateral triangle. 364. A right triangle having one leg equal to a given length. 365. A rectangle having one side equal to a given length. To construct a square equivalent to: 366. Five-eighths of a given square. 367. Three-fifths of a given pentagon. 368. To draw a line through the vertex of a given triangle so as to divide the triangle into two triangles which shall be to each other as 2:3. 369. To divide a given triangle into two equivalent parts by drawing a line through a given point P in one of the sides. 370. To find a point within a triangle, such that the lines joining this point to the vertices shall divide the triangle into three equivalent parts. 371. To divide a given triangle into two equivalent parts by drawing a line parallel to one of the sides. 372. To divide a given triangle into two equivalent parts by drawing a line perpendicular to one of the sides. 373. To divide a given parallelogram into two equivalent parts by drawing a line through a given point in one of the sides. 374. To divide a given trapezoid into two equivalent parts by drawing a line parallel to the bases. 375. To divide a given trapezoid into two equivalent parts by drawing a line through a given point in one of the bases. qual BOOK V. REGULAR POLYGONS AND CIRCLES. 395. A regular polygon is a polygon which is equilateral and equiangular; as, for example, the equilateral triangle, and the square. PROPOSITION I. THEOREM. 396. An equilateral polygon inscribed in a circle is a regular polygon. Let ABC, etc., be an equilateral polygon inscribed in a circle. To prove the polygon ABC, etc., regular. Proof. The arcs AB, BC, CD, etc., are equal, $ 230 Hence arcs ABC, BCD, etc., are equal, Ax. 6 and the A, B, C, etc., are equal, (being inscribed in equal segments). Therefore the polygon ABC, etc., is a regular polygon, being equilateral and equiangular. Q. E. D. PROPOSITION II. THEOREM. 397. A circle may be circumscribed about, and a circle may be inscribed in, any regular polygon. E B Let ABCDE be a regular polygon. I. To prove that a circle may be circumscribed about ABCDE. Proof. Let O be the centre of the circle passing through A, B, C. Join OA, OB, OC, and OD. Since the polygon is equiangular, and the ▲ OBC is isosceles, Therefore the circle passing through A, B, and C, also passes through D. |