18. If in a right triangle ABC the altitude AD is drawn to the hypotenuse, AD: AB:: AC: BC. 19. If from any point E in the chord AB, EC is drawn perpendicular to the diameter AD, ACX AD AB X AE. 20. If AB is a diameter, BD the tangent at B, and DA meets the circumference at E, AB2 = AEX AD. 21. Two triangles are similar if two sides and the median to one of these sides of the one are in proportion with the corresponding parts of the other. = 22. Construct a triangle similar to a given triangle, having given its perimeter. 23. If one chord bisects another chord, either part of the second chord is a mean proportional between the parts of the first chord. 24. Draw a line through a given point so that it will cut off on the sides of a given angle parts having a given ratio. 25. Draw a line from a given point to a given line so that it will have a given ratio to the perpendicular from that point. 26. Draw a line through a given point and between the sides of a given angle so that it will be cut in a given ratio by the point. 27. Construct two lines, having given their sum and their ratio. 28. Construct two lines, having given their difference and their ratio. 29. Construct a circumference equal to the sum of two given circumferences. 30. Construct a circumference equal to the difference of two given circumferences. 31. Inscribe a square in a semicircle. 32. Inscribe a square in a triangle. 33. State and prove the converse of Art. 249. 34. The arms of a right triangle are 24 and 32; find the hypotenuse. 35. The side of a square is 12 ft. What is the diagonal? 36. The base of an isosceles triangle is 16 and the altitude is 15. Find the sides. 37. The tangent to a circle from an outside point is 12 in. and the radius is 5 in. What is the length of the line from the outside point to the center? 38. The distance of a chord 14 in. long from the enter is 12 in. Find the radius. 39. The radius of a circle is 13 ft. What is the length of a chord 5 ft. from the center? 40. The arms of a right triangle are 8 and 15. Compute the hypotenuse and the altitude upon the hypotenuse. 41. The hypotenuse of a right triangle is 4 ft. 4 in. and one arm is 1 ft. 8 in. Find the other arm. 42. The base of an isosceles triangle is 90 ft. and the equal sides are each 53 ft. Find the altitude. 43. The radius of a circle is 9 ft. 2 in. Find the length of the tangent from a point 12 ft. 2 in. from the center. 44. The distance of a chord 24 in. long from the center is 5 in. Find the distance of a chord 10 in. long from the center. 45. Each side of an equilateral triangle is 8 ft. Find the altitude. 46. The altitude of an equilateral triangle is 16 in. Find a side. side. Solve the right triangle. NOTE: Let 2x = 47. If two sides of a triangle are 5 yds. and 8 yds. and the included angle is 60°, what is the third side? (Draw altitude to side 5. Art. 120.) 48. If two sides of a triangle are 7 and 8 and the included angle is 120°, what is the third side? 49. Find the altitude of an isosceles triangle whose base is 8 in. and whose side is 5 in. 50. Find the side of an isosceles right triangle whose base is 8. 51. If the length of the common chord of two intersecting circles is 16 and their radii are 10 and 17, what is the distance between their centers? 52. The line of centers of two circles is 10 and their radii are 4 and 13. Find the length of the common chord. 53. The diagonal of a rectangle is 41 and one side is 40. Find the other side. 54. The diagonal of a square is 20 in. Find the side. 55. The radius of a circle is 10, and the length of a tangent is 26. What is the distance from the end of the tangent to the center. 56. The diameters of two concentric circles are 22 in. and 122 in. Find the length of a chord of the larger which is tangent to the smaller. 57. The lower ends of a post and a flag pole are 42 ft. apart; the post is 8 ft. high and the pole is 48 ft. What is the length of a rope connecting their tops? 58. The radii of two circles are 16 and 34 in., and the line of centers is 82. Find the length their common external tangent; and common internal tangent. their 59. From a point 24 ft. above sea-level, one can see about 6 miles. Find the diameter of the earth. 60. A ladder 32 ft. 6 in. long is placed so that it just reaches a window 26 ft. above the street; and when turned about its foot, just reaches a window 16 ft. 6 in. from the street on the other side. Find the width of the street. 61. Find the length of the diagonal of a square whose side is 5 units. 62. One of the non-parallel sides of a trapezoid is perpendicular to the bases. If the length of this side is 40, and of the bases 31 and 22, what is the length of the other side? 63. The non-parallel sides of a trapezoid are each 25 in. in length, and one base is 40 in. longer than the other base. Find the altitude. 64. Two secants are drawn to a circle from an outside point. If their external parts are 24 and 18, while the internal part of the first is 16, what is the internal part of the second? 65. Two parallel chords on opposite sides of the center of a circle are 48 and 14, and distance between their middle points is 31. What is the radius? 66. Also find the length of a chord parallel to the given chords and midway between them. Find the distance between the chords if they are on the same side of the center. 67. If in triangle ABC, ▲ A is a right angle, and AD is the altitude upon the hypotenuse BC, AB 15, AD 9; find BD, BC, and AC. 68. AD = 8, DC = 4; find AC, BC, and AB. 69. AB 2, DC = 3; find BC, BD, and AD. 70. AB = 9, 12; AC = find BC, BD, and AD. 71. BD = 6, DC 24; find AB, AD, and AC. = = = 72. The line joining the mid-point of a chord to the mid-point of its arc is 5 in. If the chord is 24 in. long, what is the diameter? 73. If the chord of an arc is 30 and the chord of its half is 17, what is the diameter? 74. To a circle whose radius is 10, two tangents are drawn from a point, each 24 units long. Find the length of the chord joining the points of contact. 75. The parts of one chord made by a second chord are 4 and 27. The first part of the second chord is 6, what is the other part? 76. One of two intersecting chords is 38 in. long, and the parts of the other are 10 in. and 24 in. Find the parts of the first chord. 77. Find the product of the parts of any chord drawn through a point 9 units from the center of a circle whose diameter is 24 units. 78. A tangent to a circle is 12, and the secant from the same point is 18. Find the internal part of the secant. 79. The internal part of a secant 50 in. long is 32 in. Find the tangent from the same point. AREAS CHAPTER XXI AREAS OF SIMPLE FIGURES The basic method is: 300. The area of a rectangle equals the product of the base and the altitude. 301. The area of a square equals the square of its side. *302. The area of a parallelogram equals the product of the base and the altitude. *303. The area of a triangle equals one-half the product of the base and the altitude. 304. The area of a trapezoid equals one-half the sum of the bases multiplied by the altitude. 305. State a method of finding the area of a polygon. *306. The area of a regular polygon equals one-half the product of the perimeter and the apothem. *307. The area of a circle equals one-half the product of the circumference and the radius. 308. Derive the formula for the area of a circle in terms of π. 309. The area of a sector equals one-half the product of its arc and the radius. FORMULAS = 310. Let h = altitude: b base; b1 upper base; m = mid-line of trapezoid; P = perimeter; r = apothem; C = circumference; R = radius. State the formula for the area of: a. A parallelogram. b. A triangle. c. A trapezoid. d. A regular polygon. e. A circle. f. An equilateral triangle. = 311. EXERCISES 1. The area of a rhombus equals one-half the product of its diagonals. 2. The area of a triangle equals one-half the product of the perimeter and the radius of the inscribed circle. 3. The area of any circumscribed polygon equals one-half the product of its perimeter and the radius of the circle. 4. If the perimeter of a rectangle is 60 units, and the length is twice the width, find the area. 5. A canal is 14 ft. deep, 60 ft. wide at the top, and 45 ft. wide at the bottom. Find the area of a cross-section. 6. A rhombus co ns 100 sq. ft., and the length of one diagonal is 20 ft. Find the other diagonal. 7. Find the area of a right triangle, if the hypotenuse is 34, and one side is 16. high to the ridgepole. Find the entire surface area. 10. Find the area of a circle with radius 4 ft. 8. Find the area of an equilateral triangle, if one side is 8. 9. A house is 80 ft. long, 60 ft. wide, 50 ft. high to the eaves, 70 ft. 11. The area of a circle is 196 sq. in. Find the radius. 12. Find the area of a sector whose central angle is 40°, and radius 6 in. 13. The area of an equilateral triangle with side a equals 15. The area of an isosceles right triangle equals hypotenuse. 14. The area of a right triangle with one angle 60° equals where a is the hypotenuse. 4 a2 4 √ 3. where a is the |