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. 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 ACA B and ̊C A' B' AB= A'B', = also, A and B A' and B' respectively, (being alt.-int. 4), .. ACABA CA'B' ; .. CA and CBC A' and C B' respectively, (being homologous sides of equal ▲). Hyp. $ 68 $107 Now in the AACH and A' CH' A CA' C, = A and ACH A' and A' CH' respectively, ..AACH = AACH', $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. 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 Draw IKLI to X X', and I MNL to Y Y'. Now But KI=0M, (the figure being symmetrical with respect to X X'). (lls comprehended between Ils are egual). .. KL = OM. ..KLOM is a ☐, (having two sides equal and parallel). .. LO is equal and parallel to KM, (being opposite sides of a ). § 420 $ 135 Ax. 1 § 136 § 134 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 O to KM. .. 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. (12 + c2 — a2)2 2 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 cX p = √(b+c+a) (b+c− a) (a + b −c) (a−b+c), · 4 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? BOOK VI. PLANES AND SOLID ANGLES. ON LINES AND PLANES. 428. DEF. A Plane has already been defined as a surface such that the straight line joining any two points in it lies wholly in the surface. The plane is considered to be indefinite in extent, so that however far the straight line be produced, all its points lie in the plane. A plane is usually represented by a quadrilateral supposed to lie in the plane. 429. DEF. The Foot of a line is the point in which it meets the plane. 430. DEF. A straight line is perpendicular to a plane if it be perpendicular to every straight line of the plane drawn through its foot. In this case the plane is perpendicular to the line. 431. DEF. The Distance from a point to a plane is the perpendicular distance from the point to the plane. 432. DEF. A line is parallel to a plane if all its points be equally distant from the plane. In this case the plane is parallel to the line. 433. DEF. A line is oblique to a plane if it be neither perpendicular nor parallel to the plane. 434. DEF. Two planes are parallel if all the points of either be equally distant from the other. 435. DEF. The Projection of a point on a plane is the foot of the perpendicular from the point to the plane. 436. DEF. The projection of a line on a plane is the locus of the projections of all its points. 437. DEF. The plane embracing the perpendiculars which project the points of a straight line upon a plane is called the projecting plane of the line. 438. DEF. The angle which a line makes with a plane is the angle which it makes with its projection on the plane. This angle is called the Inclination of the line to the plane. 439. DEF. A plane is determined by lines or points, if no other plane can embrace these lines or points without being coincident with that plane. 440. DEF. The intersection of two planes is the locus of all the points common to the two planes. 441. An infinite number of planes may embrace the same straight line. Thus, if the plane M N embrace the line AB it may be made to revolve about A B as an axis, and to occupy an infinite number of positions, each of which is the position of a plane embracing the line A B. A M N B 442. A plane is determined by a straight line and a point without that line. Now if the plane revolve either way about the line A B as an axis, it will cease to embrace the point C. Hence any other plane embracing the line AB and the point C must be coincident with the first plane. $439 443. Three points not in a straight line determine a plane. For, by joining any two of the points, we have a straight line and a point which determine a plane. $ 442 444. Two intersecting straight lines determine a plane. For, a plane embracing one of these straight lines and any point of the other line (except the point of intersection) is determined. 445. Two parallel straight lines determine a plane. § 442 For, a plane embracing either of these parallels and any point in the other is determined. $ 442 |