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 X P Y Y B' A C 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 axis divides the figure into two sym B metrical figures. Thus, ABCDEF is symmetrical x A with respect to the axis XY, if XY divides ABCDEF into two sym 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 respect to the center O, if O bisects the line PP'. 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 symmet rical point P' by turning the radius OP about O as a pivot through 180°. A B' 506 Two figures are symmetrical with respect to a center, when every point of either has its symmetrical point in the other. Thus, AB and A'B' are symmetrical with respect to the center O, if every B point in AB has its symmetrical point in A'B'. A' about the center of symmetry as a pivot through 180°. Α' DI 507 A figure is symmetrical with respect to a center, when every line drawn through the center cuts the perimeter in two points symmetri cal with respect to the center. 508 The straight line drawn through the center of a symmetrical figure, and terminated by the perimeter, is called a diameter. PROPOSITION VIII. THEOREM 509 If a figure is symmetrical with respect to two axes perpendicular to each other, it is also symmetrical with respect to their intersection as a center. HYPOTHESIS. The figure ABCDEFGH is symmetrical with respect to the perpendicular axes XX' and YY' intersecting at O. CONCLUSION. ABCDEFGH is symmetrical with respect to 0. PROOF Let P be any point in the perimeter. Draw PLQ parallel to XX', and QKP' parallel to YY'. Join LK, OP, and OP'. Likewise POKL is a , and PO is = and || to LK. ... PO=OP', and POP' is a straight line. Ax. 11 Ax. 16 That is, the figure is symmetrical with respect to O as a center. EXERCISES § 507 Q. E. D. 1116 A circle is symmetrical with respect to its center, or with respect to any diameter as an axis. 1117 A parallelogram is symmetrical with respect to the intersection of its diagonals as a center. 1118 Every regular polygon of an even number of sides has a center of symmetry. 1119 An isosceles triangle is symmetrical with respect to the altitude upon the base. 1120 An equilateral triangle is symmetrical with respect to all of its altitudes. 1121 The symmetrical of a straight line, with respect to an axis, or with respect to a center, is an equal straight line. 1122 The symmetrical of an angle with respect to a center is an equal angle. 1123 The symmetrical of an angle with respect to an axis is an equal angle. 1124 If two straight lines are symmetrical with respect to a center, they are equal, parallel, and extend in opposite directions. 1125 If two polygons are symmetrical with respect to a center, or with respect to an axis, they are equal. 1126 A trapezium has no center of symmetry. 1127 An isosceles trapezoid has an axis of symmetry. 1128 Each diagonal of a square is an axis of symmetry. 1129 How many axes of symmetry has each of the regular polygons considered in this book ? 1130 How many axes of symmetry has a regular polygon of 2 n sides? of 2 n + 1 sides? |