Find the equation of the line satisfying the given conditions, giving it in slope–intercep...

Find the equation of the line satisfying the given conditions, giving it in slope–intercept form if possible.

Refer to the given figure and complete parts (a)–(h) to prove that if two lines are perpendicular and neither line is parallel to an axis, then the lines have slopes whose product is −1.

(a) In triangle OPQ, angle POQ is a right angle if and only if

[d(O, P)]2 + [d(O, Q)]2 = [d(P, Q)]2.

What theorem from geometry is this?

(b) Find an expression for the distance d(O, P).

(c) Find an expression for the distance d(O, Q).

(d) Find an expression for the distance d(P, Q).

(e) Use your results from parts (b)–(d) and substitute into the equation in part (a). Simplify to show that the resulting equation is −2m1m2x1x2 − 2x1x2 = 0.

(f) Factor −2x1x2 from the final form of the equation in part (e).

(g) Use the zero-product property from intermediate algebra to solve the equation in part (f) to show that m1m2 = −1.

(h) State your conclusion on the basis of parts (a)–(g).