Prove that if two nonvertical lines have slopes whose product is−1, then the lines are per...

Prove that if two nonvertical lines have slopes whose product is−1, then the lines are perpendicular. [Hint Refer to Figure and use the converse of the Pythagorean Theorem.]

Figure

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. The point A = (1, m2) is on the line having slope m2, and the point B = (1, m1) is on the line having slope m1. (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) ]2 + [d(O, B) ]2 = [d(A, B) ]2 (5)

Using the distance formula, the squares of these distances are

[d(O, A) ]2 = (1 −0)2 + (m2 − 0)2 = 1 +

[d(O, B) ]2 = (1−0)2 + (m1−0)2 = 1 +

[d(A, B)]2 = (1 − 1)2 + (m2 − m1)2 = − 2m1 m2 +

Using these facts in equation (5), we get

(1 + )+ (1 + ) = −2m1 m2 +

which, upon simplification, can be written as

m1 m2 = −1

If the lines are perpendicular, the product of their slopes is −1.