425. Two equal and parallel lines are symmetrical with respect to a centre. Let A B and A'B' be equal and parallel lines. We are to prove A B and A' B' symmetrical. Draw A A' and B B', and through the point of their intersection C, draw any other line HCH', terminated in AB and A' B'. In the AC A B and C A'B' AB= A'B', also, A and B = A' and B' respectively, .. CA and CB = CA' and C B' respectively, Now in the AACH and A' CH' Hyp. § 68 § 107 (having a side and two adj. ▲ of the one equal respectively to a side and two .. every point in A B has its symmetrical 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', YY' which intersect at 0. We are to prove O the centre of symmetry of the figure. Draw IKLI to X X', and I M N 1 to Y Y'. Now But (the figure being symmetrical with respect to X X'). KI=0M, (lls comprehended between Ils are egual). § 420 § 135 Ax. 1 $136 .. LO is equal and parallel to K M, § 134 .. KLOM is a, (having two sides equal and parallel). (being opposite sides of a □). In like manner we may prove O N equal and parallel to K M. Hence the points L, O, and N are in the same straight line drawn through the point Ol to K M. .. any straight line LO N, drawn through O, is bisected at 0. .. O is the centre of symmetry of the figure. $424 Q. E. D. EXERCISES. 1. The area of any triangle may 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. p= 4 c2 √ 4 b2 c2 - (b2 + c2 — a2)2 2 c √ (b+c+a) (b + c − a) (a + b − c) (a − b + c) 2 c Hence, show that area of ▲ A B C, which is equal to 1 √ (b+c+ a) (b+c− a) (a + b −c) (a−b+c), 2. Show that the area of an equilateral triangle, each side of 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? |