CHAPTER IV. Bisection of Lines and Angles. Perpendiculars. With A and B as 1. To bisect a straight line. Suppose AB the line to be bisected. centres describe portions of circles with equal radii intersecting at C, and with the same centres describe portions of circles with equal radii, intersecting at D. Then if CD be drawn, it bisects AB at right angles. For, using the dividers, it will be found that AE and EB are equal; and, using the protractor or set-square, all the angles at E will be found to be 90°. Or again, we would conclude that AE and EB are equal, and that the angles at E are right angles, from the symmetry of the figure with respect B to the line CD-the figure on one side of this line being just the same as the figure on the other side, but turned in the opposite direction. Or again, we may "reason out" the equality of AE and EB, and that the angles at E are right angles, as follows: Since the triangles ACD, BCD have their sides equal, they are equal in all respects (Ch. III., 1). Hence the angles at C are equal; also the sides about these angles, AC, CE, and BC, CE, are equal; therefore (Ch. III., 2) the triangles ACE and BCE are equal in all respects. Hence AE is equal to BE; also the angle AEC is equal to the angle BEC; therefore each is 90°. In practice it is not necessary to draw the lines AC, BC, AD, BD, CD. Having found the points C and D, placing the ruler on these points, we may mark the point E in AB. Subsequently, when the subject of parallel lines comes to be dealt with, another and possibly readier way of finding the middle point of a line will be given. A number of exercises should now be given in bisecting lines of different lengths, the dividers being used in each case to determine whether the point reached is the middle point. It is suggested that the pupil be given exercises in estimating with the eye the middle points of lines of various lengths, these points being afterwards accurately determined by geometrical construction. 2. To bisect an angle. Place one of Let BAC be the angle. the points of the dividers or compasses at A, and mark off equal lengths AD, AE in AB and AC. With centres D and E describe portions of circles with equal radii, intersecting at F. Then drawing AF, the angle is bisected by it. E For, adjusting the bevel to either of the angles at A, it will be found equal to the other. Or again, we would conclude that the angles at A are equal from the symmetry of the figure with respect to the line AF-the figure on one side of this line being just the same as the figure on the other side, but turned in the opposite direction. Or again, we may prove the equality of the angles as follows: The triangles DAF, EAF have their sides Hence (Ch. III., 1) the angles DAF, EAF are equal. equal. In practice it is not necessary to draw the lines DF, EF. A number of exercises should be given in bisecting angles of various magnitudes, the bevel being used in each case to determine whether the bisection is accurate. The protractor may also be used for bisecting angles. It is suggested that the pupil be given exercises in estimating with the eye the bisecting lines of a number of angles, the bisection being afterwards accurately reached by geometrical construction. Greater accuracy is likely to be secured in bisecting an angle, by making AD, AE and DF, EF of considerable length. The point F is then remote from A, and any trifling error in locating the exact point where the circles intersect, has less effect on the angle at A through being on the circumference of a large circle (radius AF). 3. From a point in a line to draw a line at right angles to it. If C be the point in AB from which the perpendicular is to be drawn, place one point of the dividers or compasses at C, and mark off equal lengths CD and CE. Then with centres D and E describe por tions of circles with equal radii, intersecting at F. Draw FC: it is perpendicular to AB. For, applying the set-square or protractor, the angles at C will be found to be right angles. Or again, from the symmetry of the figure with respect to CF, we may conclude that the angles at C are right angles. Or again, since the sides of the triangles DCF, ECF are equal, therefore (Ch. III., 1) these triangles are equal in all respects, and the angles at C are equal. Hence the angles at C are right angles. In practice the lines FD and FE need not be drawn. A number of exercises should be given in drawing lines at right angles to others from points in the latter, the correctness of the constructions being tested by using the set-square or protractor. In future, in the various constructions that are to be made, where a line is to be drawn at right angles to another from a point in the latter, the set-square or protractor should in general be used instead of this construction. 4. To draw a line perpendicular to another from a point without the latter. Let C be the point without AB from which the perpendicular is to be drawn to AB. With C as centre describe a circle cutting AB in D and E. With D and E as centres describe portions of circles with equal radii, intersecting at F. Join CF, cutting AB in G. CG is the perpendicular from C on AB. For, applying the set-square or protractor, the angles at G will be found to be right angles. Or again, from the symmetry of the figure with respect to CF, we may conclude that the angles at G are right angles. Or again, since the sides of the triangles DCF, ECF are equal, therefore, (Ch. III., 1) the angles at C are equal. Also since in the triangles DCG, ECG the angles at C are equal, and the sides about these angles equal, therefore (Ch. III., 2) these triangles are equal in all respects, and the angles at G are equal. Hence the angles at G are right angles. In practice the lines CD, CE, FD, FE, GF need not be drawn. A number of exercises should be given in drawing lines perpendicular to others from points without the latter, the correctness of the constructions being tested by using the set-square or protractor. 5. In future, where a line is to be drawn perpendicular to another from a point without the latter, the set-square or protractor should in general be used instead of the preceding construction. When for this purpose the protractor is used, the edge of the ruler is to be placed over the centre-point of the protractor and over the 90° mark; the base of the protractor is then to be slid along the line until the edge of the ruler is over the given point without the line. The centre-point of the protractor then marks the foot of the perpendicular on the line. |