42 CHAPTER 1 Graphs Figure 68 Perpendicular lines y 905 x THEOREM Criterion for Perpendicular Lines Two nonvertical lines are perpendicular if and only if the product of their slopes is −1. If two nonvertical lines are perpendicular, then the product of their slopes is −1. Figure 69 x y Run = 1 O Rise = m1 Rise = m2 A = (1, m2) B = (1, m1) Slope m2 Slope m1 1 If two nonvertical lines have slopes whose product is −1, then the lines are perpendicular. Finding the Slope of a Line Perpendicular to Another Line If a line has slope 3 2 , any line having slope − 2 3 is perpendicular to it. EXAMPLE 11 Here we prove the “only if” part of the statement: Proof Let m1 and m2 denote the slopes of the two lines.There is no loss in generality (that is, neither the angle nor the slopes are affected) if we situate the lines so that they meet at the origin. See Figure 69. The point ( ) = A m 1, 2 is on the line having slope m ,2 and the point ( ) = B m 1, 1 is on the line having slope m .1 (Do you see why this must be true?) Suppose that the lines are perpendicular. Then triangle OAB is a right triangle. As a result of the Pythagorean Theorem, it follows that ( ) [ ] ( ) [ ] ( ) [ ] + = d O A d O B d A B , , , 2 2 2 (5) Using the distance formula, the squares of these distances are d O A m m d O B m m d A B m m m mm m , 1 0 0 1 , 1 0 0 1 , 1 1 2 2 2 2 2 2 2 2 2 1 2 1 2 2 2 2 1 2 2 2 1 2 1 2 ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) [ ] ( ) = − + − = + = − + − = + = − + − = − + Using these facts in equation (5), we get m m m mm m 1 1 2 2 2 1 2 2 2 1 2 1 2 ( ) ( ) + + + = − + which, upon simplification, can be written as mm 1 1 2 = − If the lines are perpendicular, the product of their slopes is −1. ■ In Problem 138, you are asked to prove the “if” part of the theorem: You may find it easier to remember the condition for two nonvertical lines to be perpendicular by observing that the equality mm 1 1 2 = − means that m1 and m2 are negative reciprocals of each other; that is, m m m m 1 and 1 1 2 2 1 = − = − 9 Find Equations of Perpendicular Lines When two lines intersect at a right angle (90°), they are perpendicular . See Figure 68. The following result gives a condition, in terms of their slopes, for two lines to be perpendicular.
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