496 Of all isoperimetric regular polygons, that which has the greatest number of sides is the maxi The equilateral triangle ABC and the square S have equal HYPOTHESIS. perimeters. CONCLUSION. S> ABC. PROOF Join C to any point K in AB, and construct the isosceles triangle HKC isoperimetric with AKC. Then the quadrilateral HKBC is isoperimetric with the triangle ABC, and therefore isoperimetric with S. Now the ▲ HKC > the ▲ AKC. .. the quadrilateral HKBC > the ▲ ABC. § 490 Ax. 2 But S> the quadrilateral HKBC. § 495 S> ABC. In like manner it may be shown that a regular pentagon isoperimetric with S is greater than S, and so on. Q. E. D. 497 COROLLARY. The area of a circle is greater than the area of any polygon of equal perimeter. PROPOSITION VII. THEOREM 498 Of regular polygons which have the same area, that which has the greatest number of sides has the least perimeter. HYPOTHESIS. A and B are regular polygons of the same area, A having more sides than B. CONCLUSION. The perimeter of A < the perimeter of B. PROOF Construct the regular polygon C having the same perimeter as A and the same number of sides as B. Then C < A. "Of all isoperimetric regular polygons, that which has the greatest number of sides is the maximum." But AB. .. C < B. .. the perimeter of C < the perimeter of B. But the perimeter of C = the perimeter of A. .. the perimeter of A< the perimeter of B. § 496 Hyp. Const. Q. E. D. 499 COROLLARY. The circumference of a circle is less than the perimeter of any equivalent polygon. EXERCISES THEOREMS 1101 Of all triangles having the same base and the same altitude, the isosceles has the minimum perimeter. 1102 Of all triangles having the same base and the same area, the isosceles has the greatest vertical angle. 1103 Of all triangles having the same base and the same area, the isosceles has the minimum perimeter. 1104 Of all equivalent triangles, the equilateral has the minimum perimeter. 1105 Of all triangles inscribed in a given circle, the equilateral is the maximum. 1106 Of all equivalent parallelograms having equal bases, the rectangle has the minimum perimeter. 1107 Of all equivalent right parallelograms, the square has the minimum perimeter. 1108 Of all polygons of the same number of sides inscribed in a given circle, the regular polygon is the maximum. 1109 Of all triangles having the same base and equal vertical angles, the isosceles is the maximum. PROBLEMS 1110 To divide a given line into two parts so that their product shall be a maximum. 1111 To inscribe an angle in a semicircle so that the sum of its sides shall be a maximum. 1112 To draw a line through a given point within an angle so as to form the minimum triangle. 1113 To find a point in a given line so that the sum of its distances from two given points on the same side of the given line is a minimum. 1114 To inscribe in a given semicircle the maximum rectangle. 1115 To inscribe in a given semicircle the maximum trapezoid. SYMMETRY I SYMMETRY WITH RESPECT TO AN AXIS DEFINITIONS 500 Two points are symmetrical with respect to a straight line, called the axis of symmetry, when this axis bisects at right angles the straight line joining the two points. Thus, P and P' are symmetrical with respect to XY, if XY bisects PP' at right angles. 501 Two figures are symmetrical with respect to an axis, when every point in one has its symmetrical point in the other. Thus, the lines AB and A'B' are symmetrical with respect to the axis XY, if every point in either has its symmetrical point in the other, with respect to XY as the axis of symmetry. Also, the triangles ABC and A'B'C' are symmetrical with respect to the axis XY, if every X P PI B -Y point in the perimeter of one has its symmetrical point in the perimeter of the other with respect to XY as an axis. 502 A figure is symmetrical with respect to an axis, if the metrical figures, ABCD, AFED, with respect to XY. . 503 Symmetrical points and lines in two symmetrical figures are called homologous. In all cases, two figures which are symmetrical with respect to an axis, can be made to coincide by rotating either about the axis of symmetry. II SYMMETRY WITH RESPECT TO A CENTER 504 Two points are symmetrical with respect to a third point, called the center of symmetry, when this center bisects the line joining the two points. Thus, P and P' are symmetrical with P respect to the center O, if O bisects the line PP'. P 505 The distance of a point from the center of symmetry is called the radius of symmetry; as, OP, or OP'. A point P can be brought into coincidence with its symmetrical point P' by turning the radius OP about O as a pivot through 180°. 506 Two figures are symmetrical with respect to a center, when every point of either has its sym metrical point in the other. Thus, AB and A'B' are symmetrical with respect to the center O, if every point in AB has its symmetrical point in A'B'. A' B' Likewise the polygons ABCD and A'B'C'D' are symmetrical with respect to the center O. Any two figures symmetrical with respect to a center can be brought into coincidence by turn- B ing one of them in its own plane about the center of symmetry as a pivot through 180°. B |