area of the segment ACB is equal to the area of the of triangle AOB. Using the principle of Art. 87, .703 in. Area of sector 2X 86.128 X20 = 861.28 of the water in the 6-inch tank, use the proportion 6: v = 402 X 173:62 X 44, or v = 782.6 6 2 44 40 X 173 SYMMETRICAL FIGURES are said to be symmetrical with respect to axis of symmetry, when any perpendicular limited by the outline of the figure is bisected , referring to Fig. 97, if the perpendiculars 2. are bisected by the line mn, mn is an axis of the two figures A'B'C'D'E'A' are OW suppose that symmetrical, they It is also FIG. 97. plane of the paper be folded on the line mn, B will fall on B', C will fall on C', etc., and Superposed on and will coincide with A'B' left half of FBC DGD'C'B'F will be superposed le with the right half; therefore, if a plane led that every point on one side of the line of with a point on the other side, the line of folding mmetry and will divide the figure into two equal 147. If the figure have two axes of symmetry, as mn and pq, Fig. 98, their point of intersection O is called the center of symmetry, and the center of symmetry is the geometrical center of the figure. Every regular polygon has a center of symmetry, which may be m p D n_ FIG. 98. found by drawing lines from the vertexes to the opposite vertexes, when the polygons have an even number of sides, as 4, 6, 8, etc.; but, if the polygons have an odd number of sides, the axes of symmetry are found by drawing lines from the vertexes perpendicular to the sides opposite them. If the number of sides is even, a line drawn perpendicular to any side at its middle point will bisect the opposite side and be an axis of symmetry. Consequently, a regular polygon has as many axes of symmetry as it has sides. See (A) and (B), Fig. 99. The figure may be folded on any one of these axes and one half will coincide with the other half. An isosceles triangle and an arc, sector, or segment of a circle has but one axis of symmetry; see (D), Fig. 99. The figure whose outline is shown at (C) also has but one axis of symmetry. An ellipse has two axes of symmetry, the long and short diameters. 148. A solid has a plane of symmetry when sections equally distant from the plane of symmetry and parallel to it are equal and every point in one section has its symmetrical point in the other section. A frustrum of any cone has at least one plane of symmetry, which includes the axis of the cone and the long diameter of either base. If the frustum is that of a right cone, the bases are perpendicular to the axis, and there are any number of planes of symmetry perpendicular to the bases. If a solid has two planes of symmetry, they intersect in a line of symmetry, and if it has three planes of symmetry, one of which is at right angles to the other two, the three planes intersect in a point of symmetry, which is the center of the solid. Thus, in Fig. 95, if the plane AB is a plane of symmetry, that part of the rectangular parallelopiped in front of the plane is symmetrical to that part behind it; if the plane EF is also a plane of symmetry, that part of the solid to the right of the plane is symmetrical to that part to the left, and the two planes intersect in the line of symmetry pq. If a third plane AB, perpendicular to the other two is also a plane of symmetry, that part of the solid below the plane is symmetrical to that part above it, and the three planes intersect in the point 0, which is the center of the solid. 149. Now observe that when a body has one plane of symmetry, the plane divides the body into two equal parts, and the center of the body lies somewhere in this plane. When the body has two planes of symmetry, the center lies in both planes and, hence, lies somewhere in the line of their intersection; the two planes divide the body into four parts, and if they are perpendicular to each other, they divide the body into four equal parts. When the body has three planes of symmetry, the center lies in all three planes, which have only one common point-the point of intersection; the three planes divide the body into eight equal parts, and if the planes are perpendicular to one another, the eight parts are all equal. To find the center of a line, bisect the line and draw a right line perpendicular to the given line at the point of bisection; this line will be an axis of symmetry, provided the line is symmetrical (as in the case of a right line or circular arc); otherwise, it has no center, unless it is symmetrical with respect to a point. To be symmetrical with respect to a point, every line drawn through the point and limited by the given line or by the perimeter of the given figure must be bisected by the point, which is called a center of symmetry. Thus, the line shown in Fig. 100 at (a) is symmetrical with respect to the center O, because OA = OA', = FIG. 100. OB = OB', OC OC', etc., and O is the center of the line. For the same reason, the parallelogram at (b) is symmetrical with respect to the center O. The line shown at (c) is not symmetrical with respect to a center or to an axis, and therefore has no center. Observe that neither the line at (a) nor the parallelogram at (b) have an axis of symmetry-neither can be folded on any line so one-half can be superposed on the other; but both have a center of symmetry, which is the center of the figure. SOLIDS OF REVOLUTION 150. Center of Gravity. The center of gravity of a plane surface or section is that point at which the surface will balance. If the surface have an axis of symmetry, the center of gravity (which may be denoted by the abbreviation c. g.) will lie in that axis; and if it have two axes of symmetry, the c. g. will be their |
