In the right triangles OCE and OAH OE=OH. OC OA, being radii. Hence the triangles are equal. Therefore And CE=AH. CD=AB, being doubles of equals. Ax. 7 Q. E. D. PROPOSITION VIII. THEOREM 171. In the same circle or equal circles, the less of two chords is at the greater distance from the centre; conversely. the chord at the greater distance from the centre is the less. K and H are the middle points of AB and BC. Hence BK<BH. [Being halves of unequals.] 167 Ax. 8 SUMMARY: ED<BC; BA<BC ; BK<BH ; a<b; d>c ; OK>0H; OM>OH. SUMMARY: OM>OH; OK>0H; d>c; a<b; BK<BH; BA<BC · ED<BC. 172. Defs.—A straight line is tangent to a circle it, however far produced, it meets its circumference in but one This point is called the point of tangency. PROPOSITION IX. THEOREM 173. A straight line perpendicular to a radius at its extremity is tangent to the circle; conversely, the tangent at the extremity of a radius is perpendicular to that radius. GIVEN-AB perpendicular to the radius OP at its extremity P. The perpendicular OP is less than any other line OX from 0 to AB. [Being the shortest distance from a point to a line.] $96 Hence, OX being greater than a radius, X lies without the circumference, and P is the only point in AB on the circumference. Therefore AB is tangent to the circle. CONVERSELY: GIVEN TO PROVE AB tangent to the circle at P. OP perpendicular to AB. Q. E. D. Since AB is touched only at P, any other point in AB, as X, lies without the circumference. Hence OX is greater than a radius OP. Therefore OP, being shorter than any other line from O to AB, is perpendicular to AB. $96 Q. E. D. 174. COR. A perpendicular to a tangent at the point of tangency passes through the centre of the circle. Hint.-Suppose a radius to be drawn to the point of tangency. 175. CONSTRUCTION. At a point P in the circumference of a circle to draw a tangent to the circle. Draw the radius OP, and erect PB perpendicular to this radius at P. By $ 173 PB is the tangent required. 176. Exercise.-The two tangents to a circle from an exterior point are equal. Hint.-Join the given point and the centre; draw radii to points of tan gency. PROPOSITION X. THEOREM 177. If two circumferences intersect, the straight line joining their centres bisects their common chord at right angles. Ο B GIVEN two circumferences intersecting at A and B. TO PROVE―00' joining their centres is perpendicular to AB at its middle point. O and O' are each equally distant from A and B. Therefore 00' bisects AB at right angles. $150 S104 [Two points equally distant from the extremities of a straight line determine its perpendicular bisector.] Q. E. D. MEASUREMENT 178. Def.-The ratio of one quantity to another of the same kind is the number of times the first contains the second. Thus the ratio of a yard to a foot is three (3), or more fully 179. Defs.—To measure a quantity is to find its ratic to another quantity of the same kind. The second quantity is called the unit of measure; the ratio is called the numerical measure of the first quantity. Thus we measure the length of a rope by finding the number of metres in it; if it contains 6 metres, we say the numerical measure of its length is 6, the metre being the unit of measure. 180. Remark.-If the length of one rope is 20 metres, and the length of another 5 metres, the ratio of their lengths is the number of times 5 metres is contained in 20 metres— that is, the number of times 5 is contained in 20, which is written 20. We may accordingly restate § 178 as follows: The ratio of two quantities of the same kind is the ratio of their numerical measures. 181. Defs.-Two quantities are commensurable if there exists a third quantity which is contained a whole number of times in each. The third quantity is called the common measure. Thus a yard and a mile are commensurable, each containing a foot a whole number of times, the one 3 times, the other 5280 times. Again, a yard and a rod are commensurable. The common measure is not, however, a foot, as a rod contains a foot 16 times, which is not a whole number of times. But an inch is a common measure, since the yard contains it 36 times and the rod 198 times. 182. Def.—Two quantities are incommensurable if no third quantity exists which is contained a whole number of times in each. Thus it can be proved that the circumference and diameter of a circle are incommensurable; also the side and diagonal of a square. |