AMSCO'S Geometry. New York - Rye High School

The Slopes of Perpendicular Lines 307e. What are the coordinates of the intersection of CM g and BN g?f. What are the coordinates of P, the midpoint of BC?g. Write an equation of AP gthat contains the median from A.h. Does the intersection of CM gand BN glie on AP g?i. Do the medians of this triangle intersect in one point?8-4 THE SLOPES OF PERPENDICULAR LINESLet l 1be a line whose equation is y m 1xwhere m 1is not equal to 0. Then O(0, 0)and A(1, m 1) are two points on the line.slope of l 15 m 1 2 05 m 1 2 011yA(m 1, 1)1l 1A(1, m 1)5 m 1Under a counterclockwise rotationof 90° about the origin, the image ofA(1, m 1) is A(m 1, 1). Since AOA is aright angle, OA g g' OAr.gLet l 2be the line OAr throughA(m 1, 1) and O(0, 0), and let the slopeof l 2be m 2. Then:l 21y m 1xOy m 2xxm 25 0 2 10 2 (2m 1 )5 0 1 21m 152 1 m 1We have shown that when two lines through the origin are perpendicular,the slope of one is the negative reciprocal of the slope of the other.Is the rule that we found for the slopes of perpendicular lines through theorigin true for perpendicular lines that do not intersect at the origin? We willshow this by first establishing that translations preserve slope:Theorem 8.2Under a translation, slope is preserved, that is, if line l has slope m, thenunder a translation, the image of l also has slope m.