Show that the following points taken in order form the vertices of a rhombus.

(2, −3), (6, 5), (−2, 1) and (−6, −7)

Formula used:

(2, –3), (6, 5), (–2, 1) and (–6, –7)

Let the vertices be taken as A (2, –3), B (6, 5), C (–2, 1) and D (–6, –7).

Distance of AB

⇒ AB = √ ((6 – 2)² + (5 – (–3)²))

⇒ AB = √ ((6 – 2)² + (5 + 3)²)

⇒ AB = √ ((4)² + (8)²)

⇒ AB = √ (16 + 64)

⇒ AB = √ 80

Distance of BC

⇒ BC= √ ((–2 – 6)² + (1 – 5)²)

⇒ BC = √ ((–8)² + (–4)²)

⇒ BC = √ (64 + 16)

⇒ BC = √ 80

Distance of CD

⇒ CD = √ ((–6 – (–2))² + (–7 – 1)²)

⇒ CD = √ ((–6 + 2)² + (–7 – 1)²)

⇒ CD = √ ((–4)² + (–8)²)

⇒ CD = √ (16 + 64)

⇒ CD = √80

Distance of AD

⇒ AD = √ ((–6 – (2))² + (–7 – (–3))²)

⇒ AD = √ ((–6 – 2)² + (–7 + 3)²)

⇒ AD = √ ((–8)² +(–4)²)

⇒ AD = √ (64 + 16)

⇒ AD = √ 80

Distance of AC

⇒ AC = √ ((–2 – 2)² + (1 – (–3))²)

⇒ AC = √ ((–2 – 2)² + (1 + 3)²)

⇒ AC = √ ((–4)² +(4)²)

⇒ AC = √ (16 + 16)

⇒ AC = √ 32

Distance of BD

⇒ BD = √ ((–6 – 6)² + (–7 – 5)²)

⇒ BD = √ ((–6 – 6))² + (–7 – 5)²)

⇒ BD = √ ((–12)² + (–12)²)

⇒ BD = √ (144 + 144)

⇒ BD = √ 288

AB = BC = CD = DA = √80 (That is, all the sides are equal.)

AC ≠ BD (That is, the diagonals are not equal.)

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