This exercise outlines a proof of the fact that two nonvertical lines with slopes m1 and m2 are perpendicular if and only if m1m2 = −1. In the following figure, we’ve assumed that our two nonvertical lines y = m1x and y = m2x intersect at the origin. [If they did not intersect there, we could just as well work with lines parallel to these that do intersect at (0, 0), recalling that parallel lines have the same slope.] The proof relies on the following geometric fact: if and only if (OA)2 + (OB)2 = (AB)2
(a) Verify that the coordinates of A and B are A(1, m1) and B(1, m2).
(b) Show that
(c) Use part (b) to show that the equation
OA2 + OB2 = AB2
is equivalent to m1m2 = −1.
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