This exercise provides a proof of the statement about slopes of perpendicular lines in the...

This exercise provides a proof of the statement about slopes of perpendicular lines in the box 1. First, assume that L and M are nonvertical perpendicular lines that both pass through the origin. L and M intersect the vertical line x = 1 at the points (1, k) and (1, m), respectively, as shown in the figure.

(a) Use (0, 0) and (1, k) to show that L has slope k. Use (0, 0) and (1, m) to show that M has slope m.

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(b) Use the distance formula to compute the length of each side of the right triangle with vertices (0, 0), (1, k), and (1, m).

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(c) Use part (b) and the Pythagorean Theorem to find an equation involving k, m, and various constants. Show that this equation simplifies to km = −1. This proves half of the statement.

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(d) To prove the other half, assume that km = −1, and show that L and M are perpendicular as follows. You may assume that a triangle whose sides a, b, c satisfy a2 + b2 = c2 is a right triangle with hypotenuse c. Use this fact, and do the computation in part (b) in reverse (starting with km = −1) to show that the triangle with vertices (0, 0), (1, k), and (1, m) is a right triangle, so that L and M are perpendicular.

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(e) Finally, to prove the general case when L and M do not intersect at the origin, let L1 be a line through the origin that is parallel to L, and let M1 be a line through the origin that is parallel to M. Then L and L1 have the same slope, and M and M1 have the same slope (why?). Use this fact and parts (a)–(d) to prove that L is perpendicular to M exactly when km = −1.

Box 1

Two nonvertical lines, with slopes m1 and m2, are perpendicular exactly when the product of their slopes is −1, that is,