Precalculus, Chapter 1, 1.1, Section 1.1, Problem 58

Determine the length of each side of the triangle, where $P_1 = (7,2), P_2 = (-4,0)$ and $P_3 = (4,6)$. State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. (An $\textbf{isosceles triangle}$ is one in which at least two of the sides are of equal length.








$
\begin{equation}
\begin{aligned}

P_1 P_2 =& \sqrt{(-4-7)^2 + (0-2)^2}
\\
=& \sqrt{121+4}
\\
=& \sqrt{125}
\\
=& 5 \sqrt{5}
\\
\\
P_2 P_3 =& \sqrt{[4- (-4)]^2 + (6-0)^2}
\\
=& \sqrt{64+36}
\\
=& \sqrt{100}
\\
=& 10
\\
\\
P_1 P_3 =& \sqrt{(4-7)^2 + (6-2)^2}
\\
=& \sqrt{9+16}
\\
=& \sqrt{25}
\\
=& 5

\end{aligned}
\end{equation}
$


Two sides are equal.

Thus, $\Delta P_1 P_2 P_3$ is a right triangle.