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

Given points are A(4, 3), B(6, 4), C(5, 6) and D(3, 5).

We need to prove that these are the vertices of a square.

We know that in the lengths of all sides are equal and the lengths of the diagonals are equal.

Let us find the lengths of the sides.

We know that the distance between the points (x₁, y₁) and (x₂, y₂) is .

Now,

⇒ AB = √5

⇒ BC = √5

⇒ CD = √5

⇒ DA = √5

We got AB = BC = CD = DA, this may be square (or) rhombus.

Now we find the lengths of the diagonals.

⇒ AC = √10

⇒ BD = √10

We got AC = BD.

∴ The points form a square.