18. Find the area of an equilateral triangle whose side is 12. 19. Find the area of an equilateral triangle whose altitude is 6. 20. Find the side of an equilateral triangle whose area is 81√3. 21. Find the radius of the inscribed circle and the radius of the circumscribed circle of the triangle 7, 10, 11. 22. Find the radius of the inscribed circle and the radius of the circumscribed circle of the triangle 9, 10, 11. 23. The base of a parallelogram is 1 ft. 6 in. and its altitude is 2 in. Find the area. Find the side of an equal square. 24. The area of a rectangle is 540 and its base is 36. Find its alti. tude and diagonal. 25. The base of a rectangle is 3 ft. 4 in. and its diagonal is 3 ft. 5 in. Find its area. 26. Find the altitude of an equilateral triangle whose area is 360 square units. 27. The area of a polygon is 216 and its shortest side is 4. Find the area of a similar polygon whose shortest side is 5. Find the shortest side of a similar polygon three times as large. 28. The bases of a trapezoid are 2 ft. 2 in. and 4 ft. 4 in. and the altitude is 3 ft. 3 in. Find the area. 29. The area of a trapezoid is 720 sq. ft. and its bases are 3 ft. and 4 ft. 6 in. Find the altitude. 30. The equal sides of an isosceles triangle are each 17 in. and the base is 16 in. Find the area. 31. Find the area of an isosceles right triangle whose hypotenuse is 60. 32. Find the area of a square whose diagonal is 15. 33. The diagonals of a rhombus are 6 in. and 8 in. Find the area; perimeter; and altitude. 34. The sides of two equilateral triangles are 33 and 56 respectively. Find the side of an equilateral triangle equal to their sum. 35. The side of an equilateral triangle is 8. Find the side of an equilateral triangle equal to four times the first triangle. 36. Two similar polygons have corresponding sides equal to 7 and 24. Find the corresponding side of a third similar polygon equal to their sum. 37. The bases of a trapezoid are 28 and 22 and the non-parallel sides are each 5. Find the area. 38. Find the side of a square whose area is 961. 39. The projections of the arms of a right triangle upon the hypotenuse are 4 and 9. Find the area. 40. The base of a triangle is 20 in. and its altitude is 8 in. Find the area of a triangle cut off by a line parallel to the base and 4 in. from it; also 2 in. from it. 41. Find the area of a polygon circumscribed about a circle of radius 3. The perimeter of the polygon is 20. 42. The bases of a trapezoid are 6 and 11, and its altitude is 2. If the non-parallel sides are produced till they meet, find the area of the smaller triangle. 43. Find the area of a square inscribed in a circle whose radius is 9 ft. 44. The area of an isosceles right triangle is 162 sq. in. Find its hypotenuse. (Let x equal each arm.) 45. The perimeter of a polygon is 60 and the radius of the inscribed circle 5. Find the area of the polygon. 46. The sides of a triangle are 15, 14, 13. Find the area, the altitudes, the radii of the inscribed and circumscribed circles. 47. Also for 25, 63, 74. 48. One diagonal of a rhombus is five-thirds of the other, and the difference of the diagonals is 8. Find the area. 49. A trapezoid is composed of a rhombus and an equilateral triangle. Each side of each figure is 8 ft. Find the area of the trapezoid. 50. In a triangle whose base is 40 and altitude is 24, a line is drawn parallel to the base and bisecting the triangle. Find the distance from the base to this parallel. 51. The area of a polygon is 324 and one side is 18. Find the corresponding side of a similar polygon whose area is 196. 52. The sides of a triangle are 6, 7, and 8. Find the areas of the two parts into which the triangle is divided by the bisector of the angle between 6 and 7. 53. Construct a triangle equal to a given triangle and having the same base and an angle adjoining the base. 54. Construct a parallelogram equal to a given parallelogram, having the same base and an angle adjoining the base. 55. Construct a right triangle equal to a given triangle. 56. Construct a right triangle equal to a given triangle, having the hypotenuse equal to one side of the given triangle. 57. Construct a right triangle equal to a given square. 58. Construct a square equal to a given right triangle. 59. Construct a square equal to the sum of two given right triangles. 60. Construct a triangle equal to a given triangle, having its vertex angle equal to a given angle. 61. Construct a rhombus equal to a given parallelogram. 62. Construct a parallelogram equal to a given triangle. 63. Construct a square equal to a given trapezoid. 64. Construct an isosceles right triangle equal to a given triangle. 65. Construct a square equal to a given rhombus. 66. Construct a square equal to the difference of two given parallelograms. 67. Construct a square equal to the sum of several given triangles. 68. Construct a square equal to the sum of several given polygons. 69. Construct a rectangle having a given altitude and equal to a given rectangle. 70. Construct a triangle having a given base and equal to a given triangle. 71. Construct a rectangle having a given base and equal to a given triangle. 72. Construct a triangle having a given base and equal to a given polygon. 73. Construct a square having twice the area of a given square. 74. Construct a triangle three times as large as a given triangle. 75. Construct a triangle three times as large as a given similar triangle. Four times. Five times. 76. Construct an isosceles triangle equal to a given square and having a given base. 77. Draw a line parallel to the base of a triangle forming a triangle equal to one-third the given triangle. One-fourth. One-fifth. 78. Divide a triangle into two equal parts by a line drawn from any point in a side. 79. Divide a triangle into three equal parts by lines drawn from any point in a side. 80. Bisect a triangle by a line perpendicular to a side. 81. Bisect a parallelogram by a line perpendicular to a side. 82. Find the area of a square whose perimeter is 80 ft. $3. Find the area of a regular hexagon whose side is 4 and apothem is 2 √ 3. 84. Find the area of a regular dodecagon (12 sides) whose side is 10 √2-2√3 and whose apothem is 5 √2 + √3. 85. The circumference of a circle equals 12π. Find the radius and area. 86. The radius of a circle is 20. Find the circumference and area. 87. The diameter of a circle is 50 ft. Find the circumference and area. 88. Find the radius of a circle whose area is 100 sq. miles. 89. Construct a regular hexagon equal to one-half a given regular hexagon. 90. Construct a regular octagon upon a given line as side. Decagon. Dodecagon. 91. Construct a circle equal to five times a given circle. 92. Construct a circle equal to the sum of two given circles; also equal to their difference. 93. Construct a circle equal to one-half a given circle. 94. Within a given circle construct a concentric circle equal to one-half the given circle. 95. Construct a circle equal to the area bounded by two concentric circumferences. 96. The medians of a triangle divide it into six equal triangles. 97. The carrying capacity of four pipes each 3 in. in diameter is the same, neglecting friction, as that of one pipe of what diameter? 98. The diameter which bisects a chord, whose length is 168 units, is 175 units long. Find the distances from the ends of the chord to the ends of the diameter. 99. The shadow of a yardstick perpendicular to the ground is 4 ft. long. Find the height of a tree whose shadow is 100 yards. 100. The bases of a trapezoid are 6 and 10 and the altitude is 4; the non-parallel sides are produced until they meet. Find the altitude of the larger triangle. = = 101. The diagonals of a trapezoid, whose bases are AD and BC, intersect at E. If AE 18, EC 6, BD 32, find BE and ED. 102. The sides of a triangle are 6, 9, 11. the bisector of the opposite angle. Find the parts of 6 made by = 103. The sides of a triangle are 9, 12, and 16. Find the parts of 9 made by the bisector of the opposite exterior angle. 104. If the sides of a triangle are 6, 8, and 10, and the shortest side of a similar triangle is 15, find the other sides. 105. The corresponding altitudes of two similar triangles are 3 and 5 and the base of the first is 7. What is the base of the second? 106. The perimeters of two similar polygons are 36 and 60; the shortest side of the first is 2. Find the shortest side of the second. 107. If two sides of a triangle equal 15 and 25 and the projection of 15 upon 25 is 9, what is the third side? 108. The sides of a triangle are 6, 8, and 9. Find the projection of 6 upon 8, and of 8 upon 9. 109. Two sides of a triangle are 32 and 24 units and include an angle of 60°. Find the third side. 110. Two sides of a triangle are 40 and 60 units and include an angle of 45°. Find the third side. 111. The sides of a triangle are 5, 6, and 9. Find the projection of 6 upon 5; and of 9 upon 5. 112. The sides of a triangle are 4, 13, and 15. Find the three altitudes. 113. In the previous figure, find the diameter of the circumscribed circle. 114. The sides of a triangle are 9, 10, and 17. Find the three altitudes. Find the three medians. Find the median to 8. 115. The sides of a triangle are 4, 7, and 9. 116. The sides of a triangle are 7, 8, and 9. 117. In triangle ABC, AB 8, AC 11, the median to BC equals 8. Find BC. = = 118. The sides of a triangle are 3, 6, and 7. Find angle-bisector to 7. 119. The sides of a triangle are 7, 15, and 20. Find the bisector of the smallest angle. 120. If two sides of a triangle are 9 and 12, and the diameter of the circumscribed circle is 15, find the third side. 121. Given a line one unit in length, construct a line equal to √2 units; to √5 units; to √3 units. 122. Construct a line equal to the √ 10 in; 2 √6 in. 123. Find the angle and the central angle of: a regular pentagon; regular hexagon; regular octagon; regular decagon; regular dodecagon. 124. Find the length of an arc subtended by a side of an inscribed equilateral triangle in a circle whose radius is 5; of a square; of a regular inscribed hexagon; of a regular inscribed decagon; of a regular inscribed pentadecagon. 125. If the circumference of a circle is 110 yds., what is its diameter? 126. The radius of a circle is 9. What is the radius of a second circle whose circumference is twice the first circumference? 127. If the length of a quadrant is 8 meters, what is the radius? 128. A locomotive wheel is 7 ft. in diameter. How many revolutions will it make in running a mile? |