Ex. 407. Find the area of a right triangle, if the length of the hypotenuse is 17 feet and the length of one leg is 8 feet. Ex. 408. Find the ratio of the altitudes of two equivalent triangles, if the base of one is three times that of the other. Ex. 409. The bases of a trapezoid are 8 feet and 10 feet, and the altitude is 6 feet. Find the base of the equivalent rectangle that has an equal altitude. Ex. 410. Find the area of a rhombus, if the sum of its diagonals is 12 feet, and their ratio is 3: 5. Ex. 411. Find the area of an isosceles right triangle, if the hypotenuse is 20 feet. Ex. 412. In a right triangle the hypotenuse is 13 feet, one leg is 5 feet. Find the area. Ex. 413. Find the area of an isosceles triangle, if base = b, and leg = C. Ex. 414. Find the area of an equilateral triangle, if one side = 8 feet. Ex. 415. Find the area of an equilateral triangle, if the altitude h. Ex. 416. 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. Ex. 417. The sides of a right triangle are as 3:4: 5. the hypotenuse is 12 feet. Find the area. Ex. 418. Find the area of a right triangle, if one leg tude upon the hypotenuse h. The altitude upon = a, and the alti Ex. 419. Find the area of a triangle, if the lengths of the sides are 104 feet, 111 feet, and 175 feet. Ex. 420. The area of a trapezoid is 700 square feet. 30 feet and 40 feet, respectively. Find the altitude. The bases are 87 feet, BC = 119 feet, CD: Find the area. Ex. 422. 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 that has a perimeter of 400 feet and is circumscribed about the same circle of 25 feet radius (Ex. 361) ? Ex. 423. 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. Ex. 424. Upon the diagonal of a rectangle 24 feet by 10 feet a triangle equivalent to the rectangle is constructed. What is its altitude? Ex. 425. Find the side of a square equivalent to a trapezoid whose bases are 56 feet and 44 feet, and each leg is 10 feet. Ex. 426. Through a point P in the side AB of a triangle ABC, a line is drawn parallel to BC so as to divide the triangle into two equivalent parts. Find the value of AP in terms of AB. Ex. 427. What part of a parallelogram is the triangle cut off by a line from one vertex to the middle point of one of the opposite sides? Ex. 428. 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 second polygon. Ex. 429. The base of a triangle is 32 feet, its altitude 20 feet. What is the area of the triangle cut off by a line parallel to the base at a distance of 15 feet from the base ? Ex. 430. The sides of two equilateral triangles are 3 feet and 4 feet. Find the side of an equilateral triangle equivalent to their sum. Ex. 431. If the side of one equilateral triangle is equal to the altitude of another, what is the ratio of their areas? Ex. 432. 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 the bisector of the angle formed by the first two sides. Ex. 433. 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 bases and distant 1 foot from the lower base. Ex. 434. The diagonals of a rhombus are 90 yards and 120 yards, respectively. Find the area, the length of one side, and the perpendicular distance between two parallel sides. Ex. 435. Find the number of square feet of carpet that are required to cover a triangular floor whose sides are, respectively, 26 feet, 35 feet, and 51 feet. Ex. 436. If the altitude h of a triangle is increased by a length m, how much must be taken from the base a that the area may remain the same? Ex. 437. 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. BOOK V. REGULAR POLYGONS AND CIRCLES. 429. DEF. A regular polygon is a polygon which is both equilateral and equiangular. The equilateral triangle and the square are examples. PROPOSITION I. THEOREM. 430. An equilateral polygon inscribed in a circle is a regular polygon. A B Let ABC etc. be an equilateral polygon inscribed in a circle. Proof. To prove that the polygon ABC etc. is a regular polygon. § 243 Ax. 2 Therefore, arcs CFA, DFB, etc., are equal. Ax. 3 § 289 Therefore, A, B, C, etc., are equal. Therefore, the polygon ABC etc. is a regular polygon, being equilateral and equiangular. § 429 Q. E. D. 211 PROPOSITION II. THEOREM. 431. A circle may be circumscribed about, and a circle may be inscribed in, any regular polygon. E Let ABCDE be a regular polygon. 1. To prove that a circle may be circumscribed about ABCDE. Proof. Let be the centre of the circle which may be passed through A, B, and C. § 258 .. the circle passing through A, B, C, passes through D. In like manner it may be proved that the circle passing through B, C, and D also passes through E; and so on. Therefore, the circle described from O as a centre, with a radius OA, will be circumscribed about the polygon. § 231 2. To prove that a circle may be inscribed in ABCDE. Proof. Since the sides of the regular polygon are equal chords of the circumscribed circle, they are equally distant from the centre. § 249 Therefore, the circle described from O as a centre, with the distance from 0 to a side of the polygon as a radius, will be inscribed in the polygon (§ 232). Q. E. D. 432. DEF. The radius of the circumscribed circle, OA, is called the radius of the polygon. 433. DEF. The radius of the inscribed circle, OF, is called the apothem of the polygon. 434. DEF. The common centre, O, of the circumscribed and inscribed circles is called the centre of the polygon. 435. DEF. The angle between radii drawn to the extremities of any side is called the angle at the centre of the polygon. By joining the centre to the vertices of a regular polygon, the polygon can be decomposed into as many equal isosceles triangles as it has sides. 436. COR. 1. The angle at the centre of a regular polygon is equal to four right angles divided by the number of sides of the polygon. Hence, the angles at the centre of any regular polygon are all equal. 437. COR. 2. The radius drawn to any vertex of a regular polygon bisects the angle at the vertex. 438. COR. 3. The angle at the centre of a regular polygon and an interior angle of the polygon are supplementary. For § 135 Ax. 6 |