Prove that the points (0, 0), (5, 5) and (−5, 5)

The distance $d$ between two points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ is given by the formula

$d=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}}$

In an isosceles triangle there are two sides which are equal in length.

By Pythagoras Theorem in a right-angled triangle the square of the longest side will be equal to the sum of squares of the other two sides.

Here the three points are A(0,0), B(5,5) and C(−5,5).

Let us check the length of the three sides of the triangle.

$A B=\sqrt{(0-5)^{2}+(0-5)^{2}}$

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

$=\sqrt{25+25}$

$A B=5 \sqrt{2}$

$B C=\sqrt{(5+5)^{2}+(5-5)^{2}}$

$=\sqrt{(10)^{2}+(0)^{2}}$

$=\sqrt{25+25}$

$A C=5 \sqrt{2}$

Here, we see that two sides of the triangle are equal. So the triangle formed should be an isosceles triangle.

Further it is seen that

If in a triangle the square of the longest side is equal to the sum of squares of the other two sides then the triangle has to be a right-angled triangle.

Hence proved that the triangle formed by the three given points is an.