Question

Prove the $A(-5,4),B(-1,-2),C(5,2)$ are the vertices of n isosceles right angled triangle ?

Answer 1

$A(-5,4),B(-1,-2),C(5,2)\overrightarrow{AB}=4\hat{i}-6\hat{j},\overrightarrow{BC}=6\hat{i}+4\hat{j},\overrightarrow{CA}=10\hat{i}-2\hat{j}\left|\overrightarrow{AB}\right|=\sqrt{4^2+6^2}=2\sqrt{13}\left|\overrightarrow{BC}\right|=\sqrt{6^2+4^5}=2\sqrt{13}\left|\overrightarrow{CA}\right|=\sqrt{{10}^2+2^2}=2\sqrt{26}$

$\left|\overrightarrow{AB}\right|=\left|\overrightarrow{BC}\right|\therefore \angle A=\angle C$ $\therefore$ isoscles triangle.

${\left|\overrightarrow{AB}\right|}^2+{\left|\overrightarrow{BC}\right|}^2={\left|\overrightarrow{CA}\right|}^2\Rightarrow$ right angled at $B$

$\therefore$ The triangle is right angles isoscles triangle.