SYMMETRY 419. If a point bisects the straight line joining two other points, the two points are said to be symmetrical with respect to a point, and this point is called the center of symmetry. M and N are symmetrical with respect to the center A, if A bisects the straight line MN. 420. If a straight line is the perpendicular bisector of the straight line joining two points, the points are said to be symmetrical with respect to a straight line, and this line is called the axis of symmetry. M and N are symmetrical with respect to the axis XX', if XX' is the perpendicular bisector of the straight line MN. 421. If every point of one figure has a corresponding symmetrical point in another, the two figures are said to be symmetrical with respect to a center or an axis. If every point in the figure ABC has a symmetrical point in A'B'C' with respect to O as a center, then, the figures ABC and A'B'C' are symmetrical with respect to the center O. X A M -N X M D If every point in the figure DEF has a symmetrical point in D'E'F' with respect to XX as an axis, then, the figures DEF and D'E'F' are symmetrical with respect to the axis XX'. Two plane figures that are symmetrical with respect to an axis can be applied one to the other by revolving either one about the axis; consequently they are equal, and if two figures can be made to coincide by revolving one of them about an axis through 180°, they are symmetrical with respect to the axis. 422. If a point bisects every straight line drawn through it and terminated in the boundary of a figure, the figure is said to be symmetrical with respect to a point. F E If O bisects every straight line drawn through it and terminated by the boundary of ABCDEF, then, ABCDEF is symmetrical with respect to the point O. B D 423. If a straight line divides a plane figure into two parts which are symmetrical with respect to the line, the figure is said to be symmetrical with respect to a straight line. 424. Theorem. A quadrilateral which has two adjacent sides equal and the other two sides equal, is symmetrical with respect to the diagonal joining the vertices of the angles formed by the equal sides. Data: A quadrilateral, as ABCD, having AB AD, CB = CD, and the diagonal AC. = D data, and B AB= AD, CB = CD, AC is common; Δ ΑΒC = Δ ΑDC, BAC=▲ DAC, and ▲ BCA = ≤ DCA. Why? Hence, if ADC is turned on AC as an axis, it may be made to coincide with ABC. .. § 421, ADC and ABC are symmetrical with respect to AC'; that is, § 423, ABCD is symmetrical with respect to AC. Therefore, etc. Q.E.D. Proposition XXIII 425. Theorem. If a figure is symmetrical with respect to two axes perpendicular to each other, it is symmetrical with respect to their intersection as a center. Proof. From any point in the perimeter, as P, draw PMP' ▲ XX' consequently, § 150, MP'ON is a parallelogram ; P'O is equal and parallel to MN. Similarly, OQ is equal and parallel to MN. Why? Hence, points P', 0, Q are in the same straight line P'OQ, which is bisected at 0. Why? But since P is any point in the perimeter, P'OQ is any straight line drawn through 0. Hence, § 422, ABCD-H is symmetrical with respect to o as a center. Therefore, etc. Q.E.D. Ex. 685. A segment of a circle is symmetrical with respect to the perpendicular bisector of its chord as an axis. Ex. 686. A circle is symmetrical with respect to its center or with respect to any diameter as an axis. Ex. 687. A parallelogram is symmetrical with respect to the point of intersection of its diagonals as a center. a. If they are the perimeters of regular polygons of the same number of sides, and their radii. c. If they are circumferences and their radii. b. If they are the perimeters of regular polygons of the same number of sides, and their apothems. § 387 § 387 § 394 3. Two angles are equal, a. If they are angles of a regular polygon. § 374 b. If it is equilateral and inscribed in a circle. 4. An angle is bisected, a. If it is an interior angle of a regular polygon, by the radius drawn to its vertex. 5. A polygon is regular, a. If it is equilateral and equiangular. § 382 § 374 equal divisions of the circumference of a circle. c. If it is formed by chords joining the extremities of arcs which are § 375 d. If it is formed by tangents drawn at the extremities of arcs which are equal divisions of the circumference of a circle. § 376 e. If it is formed by tangents to a circle at the. middle points of the arcs subtended by the sides of a regular inscribed polygon. § 376 § 384 6. Polygons are similar, a. If they are regular and have the same number of sides. § 386 7. A regular polygon is equivalent, a. To half the rectangle formed by its perimeter and apothem. $388 8. A circle is equivalent, a. To half the rectangle formed by its circumference and radius. § 397 9. A circumference is the limit, a. Of the perimeter of a regular inscribed polygon when the number of its sides is indefinitely increased. $ 392 b. Of the perimeter of a regular circumscribed polygon when the number of its sides is indefinitely increased. § 392 10. A circle is the limit, a. Of a regular inscribed polygon when the number of its sides is indefinitely increased. § 392 b. Of a regular circumscribed polygon when the number of its sides is indefinitely increased. 11. Figures are in proportion, § 392 a. If they are regular polygons of the same number of sides, to the squares upon their radii. § 390 b. If they are regular polygons of the same number of sides, to the squares upon their apothems. c. If they are circles, to the squares of their radii. § 390 d. If they are similar sectors, to the squares of their radii. $ 399 § 401 § 402 12. The area of a figure is equal, a. If it is a regular polygon, to one half the product of its perimeter by its apothem. § 389 b. If it is a circle, to one half the product of its circumference by its radius. § 397 § 398 d. If it is a sector, to one half the product of its arc by its radius. § 400 c. If it is a circle, to π times the square of its radius. SUPPLEMENTARY EXERCISES Ex. 688. If the perimeter of each of the figures, equilateral triangle, square, and circle is 396 ft., what is the area of each figure? Ex. 689. If the inscribed and circumscribed circles of a triangle are concentric, the triangle is equilateral. Ex. 690. If an equilateral triangle is inscribed in a circle, any side will cut off one fourth of the diameter from the opposite vertex. Ex. 691. The square inscribed in a circle is equivalent to one half the square circumscribed about that circle. Ex. 692. A circle is inscribed in a square whose side is 4 in. How much of the area of the square is without the circle ? Ex. 693. What is the width of the ring between the circumferences of two concentric circles whose circumferences are 48 ft. and 36 ft. respectively? Ex. 694. Of all squares that can be inscribed in a given square, the minimum has its vertices at the middle points of the sides. Ex. 695. Every equiangular polygon circumscribed about a circle is regular. Ex. 696. In any regular polygon of an even number of sides, the lines joining opposite vertices are diameters of the circumscribing circle. |