We used the similar triangles to show that the product of the slopes of two perpendicular lines equals -1 . The steps below outline an alternative proof that avoids the use of $\operatorname{sim}-$ ilar triangles but uses more algebra instead. Use the figure below, which is the same as the figure used earlier except that there is now no need to label the angles.
(a) Apply the Pythagorean Theorem to triangle $P S Q$ to find the length of the line segment $P Q$ in terms of $a$ and $b$.
(b) Apply the Pythagorean Theorem to triangle $P S T$ to find the length of the line segment $P T$ in terms of $a$ and $c$.
(c) Apply the Pythagorean Theorem to triangle $Q P T$ to find the length of the line segment $Q T$ in terms of the lengths of the line segments of $P Q$ and $P T$ calculated in the first two parts of this problem.
(d) As can be seen from the figure, the length of the line segment $Q T$ equals $b+c$. Thus set the formula for length of the line segment $Q T$, as calculated in the previous part of this problem, equal to $b+c,$ and solve the resulting equation for $c$ in terms of $a$ and $b$.
(e) Use the result in the previous part of this problem to show that the slope of the line containing $P$ and $Q$ times the slope of the line containing $P$ and $T$ equals -1 .