How do I prove that (-3, 0), (0,4), (3,0) are the vertices of an isosceles triangle?

Jan 5, 2016

The sides $P 1 P 2 = P 2 P 3 = 5$ but $P 3 P 1 = 3$ => ${\triangle}_{P 1 P 2 P 3}$ is isosceles

Explanation:

Find the size of the sides of the triangle and show that two sides are equal but the third one is different than the others.

$P 1 P 2 = \sqrt{{\left(0 + 3\right)}^{2} + {\left(4 - 0\right)}^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5$
$P 2 P 3 = \sqrt{{\left(3 - 0\right)}^{2} + {\left(0 - 4\right)}^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5$
$P 3 P 1 = \sqrt{{\left(- 3 - 3\right)}^{2} + {\left(0 - 0\right)}^{2}} \equiv \sqrt{9 + 0} = \sqrt{9} = 3$

The sides $P 1 P 2 = P 2 P 3 = 5$ but $P 3 P 1 = 3$ => ${\triangle}_{P 1 P 2 P 3}$ is isosceles