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

(−3, 0), (1, −2), (5, 6) and (1, 8)

Formula used:

(–3, 0), (1, –2), (5, 6) and (1, 8)

Let the vertices be taken as A (–3, 0), B (1, –2), C (5, 6) and D (1, 8).

Distance of AB

⇒ AB = √ ((1 – (–3))² + ((–2 – 0)²)

⇒ AB = √ ((1 + 3)² + (–2 – 0)²)

⇒ AB = √ ((4)² + (–2)²)

⇒ AB = √ (16 + 4)

⇒ AB = √ 20

Distance of BC

⇒ BC= √ ((5 – 1)² + (6 – (–2))²)

⇒ BC = √ ((5 – 1)² + (6 + 2)²)

⇒ BC = √ ((4)² + (8)²)

⇒ BC = √ (16 + 64)

⇒ BC = √ 80

Distance of CD

⇒ CD = √ ((1 – 5)² + (8 – 6)²)

⇒ CD = √ ((–4)² + (2)²)

⇒ CD = √ (16 + 4)

⇒ CD = √ 20

Distance of AD

⇒ AD = √ ((1 – (–3))² + (8 – 0)²)

⇒ AD = √ ((1 + 3)² + (8 – 0)²)

⇒ AD = √ ((4)² + (8)²)

⇒ AD = √ (16 + 64)

⇒ AD = √ 80

Distance of AC

⇒ AC = √ ((5 – (–3))² + (6 – 0)²)

⇒ AC = √ ((5 + 3)² + (6 – 0)²)

⇒ AC = √ ((8)² + (6)²)

⇒ AC = √ (64 + 36)

⇒ AC = √ 100

⇒ AC = 10

Distance of BD

⇒ BD = √ ((1 – 1)² + (8 – (–2))²)

⇒ BD = √ ((1 – 1)² + (8 + 2)²)

⇒ BD = √ ((0)² + (10)²)

⇒ BD = √ (0 + 100)

⇒ BD = √ 100

⇒ BD = 10

AB = CD = √20 and BC = AD = √ 80 (opposite sides of rectangle are equal).

AC = BD = 10 (Diagonals of rectangle are equal)

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