Question

Check whether $(5,-2)$, $(6,4)$ and $(7,-2)$ are the vertices of an isosceles triangles.

Answer 1

Let ABC be the triangle $A(5,-2),B(6,4),C(7,-2)$

By distance formula

$AB=\sqrt{(5-6{)}^2+(-2-4{)}^2}=\sqrt{(-1{)}^2+(-6{)}^2}\sqrt{37}$

$BC=\sqrt{(6-7{)}^2+(4-(-2{)}^2}=\sqrt{(-1{)}^2+(6{)}^2}=\sqrt{37}$

$CA=\sqrt{(7-5{)}^2+(-2-(-2){)}^2}=\sqrt{(2{)}^2+0}=2$

Since $AB=BC=\sqrt{37}$

Hence $(5,-2),(6,4)$ & $(7,-2)$ are vertices of isosceles triangle.