#### Question

Prove that (4, 3), (6, 4) (5, 6) and (3, 5) are the angular points of a square.

#### Solution

Let A (4, 3); B (6, 4); C (5, 6) and D (3, 5) be the vertices of a quadrilateral. We have to prove that the quadrilateral ABCD is a square.

So we should find the lengths of sides of quadrilateral ABCD.

`AB = sqrt((6 - 4)^2 + (4 -3)^2)`

`= sqrt(4 + 1)`

`= sqrt5`

`BC = sqrt((6 - 5)^2 + (4 - 6)^2)`

`= sqrt(1 + 4)`

`= sqrt5`

`CD = sqrt((3 - 5)^2 + (5 - 6)^2)`

`= sqrt(4 + 1)`

`= sqrt5`

`AD = sqrt((3 - 4)^2 + (5 - 3)^2)`

`= sqrt(1+ 4)`

`= sqrt5`

All the sides of quadrilateral are equal.

So now we will check the lengths of the diagonals.

`AC = sqrt((5 - 4)^2 + (6 - 3)^2)`

`=sqrt(1 + 9)`

`= sqrt(10)`

`BC = sqrt((6 - 3)^2 + (4 - 5)^2)`

`= sqrt(9 + 1)`

`= sqrt10`

All the sides as well as the diagonals are equal. Hence ABCD is a square.