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

(−1, 1), (0, 0), (3, 3) and (2, 4)

Formula used:

(–1, 1), (0, 0), (3, 3) and (2, 4)

Let the vertices be taken as A (–1, 1), B (0, 0), C (3, 3) and D (2, 4).

Distance of AB

⇒ AB = √ ((0 – (–1))² + (0 – 1)²)

⇒ AB = √ ((0 + 1)² + (0 – 1)²)

⇒ AB = √ ((1)² + (–1)²)

⇒ AB = √ (1 + 1)

⇒ AB = √ 2

Distance of BC

⇒ BC= √ ((3 – 0)² + (3 – 0)²)

⇒ BC = √ ((3)² + (3)²)

⇒ BC = √ (9 + 9)

⇒ BC = √ 18

Distance of CD

⇒ CD = √ ((2 – 3)² + (4 – 3)²)

⇒ CD = √ ((1)² + (1)²)

⇒ CD = √ (1 + 1)

⇒ CD = √2

Distance of AD

⇒ AD = √ ((2 – (–1))² + (4 – 1)²)

⇒ AD = √ ((2 + 1)² + (4 – 1)²)

⇒ AD = √ ((3)² + (3)²)

⇒ AD = √ (9 + 9)

⇒ AD = √ 18

Distance of AC

⇒ AC = √ ((3 – (–1))² + (3 – 1)²)

⇒ AC = √ ((3 + 1)² + (3 – 1)²)

⇒ AC = √ ((4)² + (2)²)

⇒ AC = √ (16 + 4)

⇒ AC = √ 20

Distance of BD

⇒ BD = √ ((2 – 0)² + (4 – 0)²)

⇒ BD = √ ((2)² + (4)²)

⇒ BD = √ (4 + 16)

⇒ BD = √ 20

AB = CD = √2 and BC = AD = √ 18 (opposite sides of rectangle are equal).

AC = BD = √ 20 (Diagonals of rectangle are equal)

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