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

(−4, −7), (−1, 2), (8, 5) and (5, −4)

Formula used:

(–4, –7), (–1, 2), (8, 5) and (5, –4)

Let the vertices be taken as A (–4,–7), B (–1, 2), C (8, 5) and D (5, –4).

Distance of AB

⇒ AB = √ ((–1 – (–4))² + (2 – (–7)²))

⇒ AB = √ ((–1+4)² + (2+7)²)

⇒ AB = √ ((3)² + (9)²)

⇒ AB = √ (9 + 81)

⇒ AB = √ 100

⇒ AB = 10

Distance of BC

⇒ BC= √ ((8 – (–1))² + (5 – 2)²)

⇒ BC = √ ((8+1)² + (3)²)

⇒ BC = √ ((9)²+ 9)

⇒ BC = √ (81 + 9)

⇒ BC = √ 100

⇒ BC = 10

Distance of CD

⇒ CD = √ ((5 – 8)² + (–4 –5 )²)

⇒ CD = √ ((3)² + (–9)²)

⇒ CD = √ (9 + 81)

⇒ CD = √100

⇒ CD = 10

Distance of AD

⇒ AD = √ ((5 – (–4))² + (–4 –(–7) )²)

⇒ AD = √ ((5+4)² + (–4+7)²)

⇒ AD = √ ((9)² +(3)²)

⇒ AD = √ (81+9)

⇒ AD = √ 100

⇒ AD = 10

Distance of AC

⇒ AC = √ ((8 – (–4))² + (5 – (–7))²)

⇒ AC = √ ((8+4)² + (5+7)²)

⇒ AC = √ ((12)² +(12)²)

⇒ AC = √ (144 + 144)

⇒ AC = √ (288)

Distance of BD

⇒ BD = √ ((5 – (–1))² + (–4 – 2)²)

⇒ BD = √ ((5 + 1))² + (–4 – 2)²)

⇒ BD = √ ((6)² + (–6)²)

⇒ BD = √ (36 + 36)

⇒ BD = √ 72

AB = BC = CD = DA = 10 (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.