Question

Prove that the points (0, 0), (5, 5) and (-5, 5) are the vertices of a right isosceles triangle.

Answer 1

The distance d between two points `(x_1, y_1)` and `(x_2, y_2)` is given by the formula

`d = sqrt((x_1- x_2)^2 + (y_1 - y_2)^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.

`AB = sqrt((0 - 5)^2 + (0 - 5)^2)`

`= sqrt((-5)^2 + (0 - 5)^2)`

`= sqrt(25 + 25)`

`AB = 5sqrt2`

`BC = sqrt((5 + 5)^2 + (5 - 5)^2)`

`= sqrt((10)^2 + (0)^2)`

`= sqrt(100)`

BC = 10

`AC = sqrt((0 + 5)^2 + (0 - 5)^2)`

`= sqrt((5)^2 + (-5)^2)`

`=sqrt(25 + 25)`

`AC =  5sqrt2`

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