# Do the points represent the vertices of a right triangle (1,-4) (5,6) (-2,3)?

Jul 27, 2017

The $\Delta$ is an isosceles right triangle, right-angled at the pt.

$\left(- 2 , 3\right) .$

#### Explanation:

Let us name the given Points (pts.) $A , B , \mathmr{and} , C ,$ resp.

We use the Notation $A B$ to denote the Distance between

the pts. $A \mathmr{and} B .$

Using the Distance Formula, we find that,

AB^2=(1-5)^2+(-4-6)^2=16+100=116;

BC^2=(5+2)^2+(6-3)^2=49+9=58; and,

$A {C}^{2} = {\left(1 + 2\right)}^{2} + {\left(- 4 - 3\right)}^{2} = 9 + 49 = 58.$

Clearly, $B C = A C , \mathmr{and} , B {C}^{2} + A {C}^{2} = A {B}^{2.}$

Therefore, we conclude that, $\Delta A B C$ is an isosceles right

triangle, right-angled at the pt. $C .$