Question

Determine \(k\) so that the points \(A(7,5), B(-1,2)\), and \(C(k, 0)\) are the vertices of a right triangle with right angle at B.

Answer 1

The condition for \(\triangle A B C\) to have a right angle at \(B\) is that lines \(\overrightarrow{A B}\) and \(\overleftrightarrow{B C}\) are perpendicular, which holds when and only when the product of their slopes is \(-1\), that is \(\left(\frac{3}{8}\right)[-2 /(1+k)]=-1\). This is equivalent to \(6=8(1+k), \quad\) or \(8 k=-2\), or \(k=-\frac{1}{4}\).