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

(8, 3), (0, −1), (−2, 3) and (6, 7)

Formula used:

(8, 3), (0, –1), (–2, 3) and (6, 7)

Let the vertices be taken as A (8, 3), B (0, –1), C (–2, 3) and D (6, 7).

Distance of AB

⇒ AB = √ ((0 – 8)² + ((–1 – 3)²)

⇒ AB = √ ((–8)² + (–4)²)

⇒ AB = √ (64 + 16)

⇒ AB = √ 80

Distance of BC

⇒ BC= √ ((–2 – 0)² + (3 – (–1))²)

⇒ BC = √ ((–2 – 0)² + (3 + 1)²)

⇒ BC = √ ((–2)² + (4)²)

⇒ BC = √ (4 + 16)

⇒ BC = √ 20

Distance of CD

⇒ CD = √ ((6 – (–2))² + (7 – 3)²)

⇒ CD = √ ((6 + 2)² + (7 – 3)²)

⇒ CD = √ ((8)² + (4)²)

⇒ CD = √ (64 + 16)

⇒ CD = √80

Distance of AD

⇒ AD = √ ((6 – 8)² + (7 – 3)²)

⇒ AD = √ ((–2)² + (4)²)

⇒ AD = √ (4 + 16)

⇒ AD = √ 20

Distance of AC

⇒ AC = √ ((–2 – 8)² + (3 – 3)²)

⇒ AC = √ ((–10)² + (0)²)

⇒ AC = √ (100 + 0)

⇒ AC = √ 100

⇒ AC = 10

Distance of BD

⇒ BD = √ ((6 – 0 )² + (7 – (–1))²)

⇒ BD = √ ((6 – 0)² + (7 + 1)²)

⇒ BD = √ ((6)² + (8)²)

⇒ BD = √ (36 + 64)

⇒ BD = √ 100

⇒ BD = 10

AB = CD = √80 and BC = AD = √ 20 (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.