Answer
As $d_1^2+d_3^2=d_2^2$, the three sides verify the Pythagorean Theorem, so the triangle is a right triangle.
Work Step by Step
We are given the points:
$(-1,3)$
$(3,5)$
$(5,1)$
Determine the length of each side using the formula
$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$,
where $(x_1,y_1),(x_2,y_2)$ are the coordinates of the two points.
$d_1=\sqrt{[3-(-1)]^2+(5-3)^2}=\sqrt{20}=2\sqrt 5$
$d_2=\sqrt{(5-3)^2+(1-5)^2}=\sqrt{20}=2\sqrt 5$
$d_3=\sqrt{[5-(-1)]^2+(1-3)^2}=\sqrt{40}=2\sqrt{10}$
Compute $d_1^2+d_2^2$:
$d_1^2+d_2^2=(2\sqrt 5)^2+(2\sqrt 5)^2=20+20=40$
Compute $d_3^2$:
$d_3^2=(2\sqrt{10})^2=40$
As $d_1^2+d_3^2=d_2^2$, the three sides verify the Pythagorean Theorem, so the triangle is a right triangle.
