Examine whether the following points taken in order form a square.

(3, 2), (0, 5), (−3, 2) and (0, −1)

Formula used:

(3, 2), (0, 5), (–3, 2) and (0, –1)

Let the vertices be taken as A (3, 2), B (0, 5), C (–3, 2) and D (0, –1).

Distance of AB

⇒ AB = √ ((0 – 3)² + ((5 – 2)²)

⇒ AB = √ ((–3)² + (3)²)

⇒ AB = √ (9 + 9)

⇒ AB = √18

Distance of BC

⇒ BC= √ ((–3 – 0)² + (2 – 5)²)

⇒ BC = √ ((–3)² + (–3)²)

⇒ BC = √ (9 + 9)

⇒ BC = √ 18

Distance of CD

⇒ CD = √ ((0 – (–3))² + (–1 – 2)²)

⇒ CD = √ ((0 + 3)² + (–1 – 2)²)

⇒ CD = √ ((3)² + (–3)²)

⇒ CD = √ (9 + 9)

⇒ CD = √18

Distance of AD

⇒ AD = √ ((0 – 3)² + (–1 – 2)²)

⇒ AD = √ ((–3)² + (–3)²)

⇒ AD = √ (9 + 9)

⇒ AD = √ 18

Distance of AC

⇒ AC = √ ((–3 – 3)² + (2 – 2)²)

⇒ AC = √ ((–6)² + (0)²)

⇒ AC = √ (36 + 0)

⇒ AC = √36

⇒ AC = 6

Distance of BD

⇒ BD = √ ((0 – 0)² + (–1 – 5)²)

⇒ BD = √ ((0)² + (–6)²)

⇒ BD = √ (0 + 36)

⇒ BD = √ 36

⇒ BD = 6

AB = BC = CD = DA = √18. (That is, all the sides are equal.)

AC = BD = 6. (That is, the diagonals are equal.)

Hence the points A, B, C and D form a square.