Precalculus, Chapter 1, 1.1, Section 1.1, Problem 56

Determine the length of each side of the triangle, where $P_1 = (-1,4), P_2 = (6,2)$ and $P_3 = (4,-5)$. 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{[6-(-1)]^2 + (2-4)^2}
\\
=& \sqrt{49+4}
\\
=& \sqrt{53}
\\
\\
P_2 P_3 =& \sqrt{(4-6)^2 + (-5-2)^2}
\\
=& \sqrt{4 + 49}
\\
=& \sqrt{53}
\\
\\
P_1 P_3 =& \sqrt{[4-(-1)]^2 + (-5-4)^2}
\\
=& \sqrt{25+81}
\\
=& \sqrt{106}

\end{aligned}
\end{equation}
$


Two sides are equal.

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