162. Divide a trapezoid into two equivalent parts by a line perpendicular to the bases. 163. Construct on AB as a base an isosceles triangle equivalent to the triangle ABC. 164. Divide a triangle into two equivalent parts by a line parallel to one side. 165. Construct a triangle in which the altitude and the base are equal and which is equivalent to a given triangle. 166. Through a given point in one side of a triangle draw a line dividing the triangle into two equivalent parts. HINT. Apply § 325 or draw the median to the side on which the given point lies. BOOK V NUMERICAL EXERCISES 167. In Query 17 on page 146 of Book II compute the area over which the horse can graze. 168. The radius of a circle is 10". Show that the difference between its area and that of the regular circumscribed hexagon is 32.25 square inches. 169. The radius of a circle is 13' and the chord of a segment is 10'. Find the area of the larger of the two segments. 170. The side of a square is 10. A circle has the same perimeter as the square. Compare the areas of the square and the circle. 171. The hands of a clock are 10′′ and 14′′ respectively. How far does the outer extremity of the minute hand move in the time from 12.40 P.M. to 7.20 P.M. (Use 22 for π.) 172. Under the same pressure how many times as much water per minute will be delivered by a pipe whose inside diameter is 6" as by one whose inside diameter is 2"? 173. A circular tank 8 feet in diameter and 40 feet high contains 250 tons of water. Find the pressure on one square inch of the bottom 174. The rim of a circular saw 4 feet in diameter runs 10,000 feet per minute. How many revolutions per second does the saw make? 175. Show that the length of the side of a regular inscribed pentagon is R B x A L H K 176. Show that the apothem of a regular inscribed pentagon is (1 +√5). 177. Show that the area of a regular inscribed pentagon is 178. The radius of a circle is 10. Show that the area of its regular inscribed pentagon is 237.76 +. 180. From a study of the preceding exercise state a method different from that of Problem 5 for inscribing a regular decagon and a regular pentagon in a circle. Addition, 164 Alternation, 161 INDEX Altitude, of parallelogram, 218; of Antecedent, 156 Arc, 89; major, 90; minor, 90 Area, 214; of a circle, 275; of irreg- 215 Axiom, 7 Bases, of parallelograms, 218; of trapezoid, 70 Bisection, 12 Boundaries, 6 Center of circle, 89; of polygon, 261 Chord, 93; common, 108 Circumscribed polygon, 103 Construction, instruments used in, Convex polygons, 64 Corresponding angles, 26 Corresponding lines, 169 Corresponding parts, 10 Decagon, 255 Definition, 3; of π, 268 Diagonal, 35 Diameter, 89 Dimensions, 4 Distance to a line, 62 Equality, 214; of magnitudes, 8 Equiangular triangle, 52 Equilateral polygon, 46 Equilateral triangle, 52 Euclid's "Elements," 86 Extending a line, 47 External common tangent, 107 Fallacies, geometric, 86 Figure, geometric, 6; plane, 6; rec- Figures, drawing of, 45-46 Geometric fallacies, 86 Hexagon, 35 Homologous, 10 Hypotenuse, 23 Incommensurable, 157 Inequalities, 72; order of, 72 Inscribed angle, 116 Inscribed polygon, 114 Intercept, 90 Internal common tangent, 107 Inversion, 164 Isosceles triangle, 13 Line of centers, 107 Line-segment, 5 Lines, 4; concurrent, 62; corre- Linkage, 233 Loci, 143 Locus, 143 Magnitudes, 4; commensurable, 157; Perpendicular, 16; mid-, 60 π, computation of, 270; definition of, Polygon, 34; angle at center of, 261; Polygons, convex, 64; equilateral, Postulates of construction, 130 Proof, 7; method of, 42-44; by re- Proportion, 158 Quadrilateral, 33 Radius, 89; of a polygon, 260 |