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422. Two plane figures are symmetrical with respect tu a centre, an axis, or a plane, if every point of either figure have its corresponding symmetrical point in the other.
Thus, the lines A B and A' B' are symmetrical with respect to the centre C (Fig. 1), to the axis XX' (Fig. 2), to the plane MN (Fig. 3), if every point of either have its corresponding symmetrical point in the other.
Also, the triangles A B D and A' B' D' are symmetrical with respect to the centre C (Fig. 4), to the axis X X' (Fig. 5), to the plane M N (Fig. 6), if every point in the perimeter of either have its corresponding symmetrical point in the perimeter of the other.
423. Def. In two symmetrical figures the corresponding symmetrical points and lines are called homologous.
Two symmetrical figures with respect to a centre can be brought into coincidence by revolving one of them in its own plane about the centre, every radius of symmetry revolving through two right angles at the same time.
Two symmetrical figures with respect to an axis can be brought into coincidence by the revolution of either about the axis until it comes into the plane of the other.
424. DEF. A single figure is a symmetrical figure, either when it can be divided by an axis, or plane, into two figures symmetrical with respect to that axis or plane; or, when it has a centre such that every straight line drawn through it cuts the perimeter of the figure in two points which are symmetrical with respect to that centre.
Fig. 2. Thus, Fig. 1 is a symmetrical figure with respect to the axis X X', if divided by XX' into figures A B C D and A B'C'D which are symmetrical with respect to X X'.
And, Fig. 2 is a symmetrical figure with respect to the centre 0, if the centre o bisect every straight line drawn · through it and terminated by the perimeter.
Every such straight line is called a diameter.
The circle is an illustration of a single figure symmetrical with respect to its centre as the centre of symmetry, or to any diameter as the axis of symmetry.
PROPOSITION XXX. THEOREM. 425. Two equal and parallel lines are symmetrical with respect to a centre.
Let A B and A' B' be equal and parallel lines.
Draw A A and B B', and through the point of their inter-
A B = A'B',
(being alt.-int. É),
§ 107 .:C A and C B= C A and C B' respectively,
(being homologous sides of equal S).
AC = A'C',
§ 107 (having a side and two adj. € of the one equal respectively to a side and two
adj. e of the other).
But H is any point in A B;
.. A B and A' B' are symmetrical with respect to C as a centre of symmetry.
Q. E. D. 426. COROLLARY. If the extremities of one line be respectively the symmetricals of another line with respect to the same centre, the two lines are symmetrical with respect to that centre.
PROPOSITION XXXI. THEOREM. 427. If a figure be symmetrical with respect to two axes perpendicular to each other, it is symmetrical with respect to their intersection as a centre.
Let the figure ABCDEFGH be symmetrical to the
two axes X X', Y Y which intersect at 0.
Join L 0, 0 N, and K M.
§ 135 (lls comprehended between Ils are egual). .. KL= 0 M.
Ax. 1 .. KLO M is a D,
§ 136 (having two sides equal and parallel). .. LO is equal and parallel to KM, $ 134
(being opposite sides of a O). In like manner we may prove O N equal and parallel to K M.
Hence the points L, 0, and N are in the same straight line drawn through the point o 1 to K M. Also
(since each is equal to KM). .. any straight line LON, drawn through 0, is bisected at 0. ..O is the centre of symmetry of the figure. § 424
Q. E. D.
EXERCISES. 1. The area of any triangle inay be found as follows: From half the sum of the three sides subtract each side severally, multiply together the half sum and the three remainders, and extract the square root of the product.
Denote the sides of the triangle A B C by a, b, c, the alti
, a+b+c tude by P, and by s. Show that a’= 12 + c2 -- 2 cXAD,
12+ 6 - a? A D= 2ci and show that
V 4 622 – (12 + 2 — a)2
с Хр Hence, show that area of A A B C, which is equal to
= 1 V (6+c+a) (b+c-a) (a +6–c) (a−6+c), =V s (s—a) (s—b) (8—c).
2. Show that the area of an equilateral triangle, each side of
a' V 3 which is denoted by a, is equal to .
3. How many acres are contained in a triangle whose sides are respectively 60, 70, and 80 chains ?
4. How many feet are contained in a triangle each side of which is 75 feet ?