Q. 7 E5.0( 1 Vote )

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

(15, 20), (−3, 12), (−11, −6) and (7, 2)

Answer :

Formula used:

(15, 20), (–3, 12), (–11, –6) and (7, 2)

Let the vertices be taken as A (15, 20), B (–3, 12), C (–11, –6) and D (7, 2).

Distance of AB

⇒ AB = √ ((–3 – 15)^{2} + (12 – 20)^{2})

⇒ AB = √ ((–18)^{2} + (–8)^{2})

⇒ AB = √ (324 + 64)

⇒ AB = √ 388

Distance of BC

⇒ BC= √ ((–11 –(–3))^{2} + (–6 – 12)^{2})

⇒ BC = √ (–11 + 3)^{2} + (–6 – 12)^{2})

⇒ BC = √ ((–8)^{2} + (–18)^{2})

⇒ BC = √ (64 + 324)

⇒ BC = √ 388

Distance of CD

⇒ CD = √ ((7 – (–11))^{2} + (2 – (–6))^{2})

⇒ CD = √ ((7 + 11)^{2} + (2 + 6)^{2})

⇒ CD = √ ((18)^{2} + (8)^{2})

⇒ CD = √ (324 + 64)

⇒ CD = √388

Distance of AD

⇒ AD = √ ((7 – 15))^{2} + (2 – 20)^{2})

⇒ AD = √ ((–8)^{2} +(–18)^{2})

⇒ AD = √ (64 + 324)

⇒ AD = √ 388

Distance of AC

⇒ AC = √ ((–11 – 15)^{2} + (–6 – 20)^{2})

⇒ AC = √ ((–26)^{2} +(–26)^{2})

⇒ AC = √ (676 + 676)

⇒ AC = √ 1352

Distance of BD

⇒ BD = √ ((7 – (–3))^{2} + (2 – 12)^{2})

⇒ BD = √ ((7 + 3))^{2} + (2 – 12)^{2})

⇒ BD = √ ((10)^{2} + (–10)^{2})

⇒ BD = √ (100 + 100)

⇒ BD = √ 200

AB = BC = CD = DA = √388 (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.

Rate this question :

Find the distance between the following pairs of points.

(2, 2) and (3, 2)

Tamilnadu Board Math Term-IShow that the following points form an equilateral triangle.

(2, 2), (−2, −2) and (−2√3, 2√3)

Tamilnadu Board Math Term-IShow that the following points form an isosceles triangle.

(−2, 0), (4, 0) and (1, 3)

Tamilnadu Board Math Term-IShow that the following points form an isosceles triangle.

(−1, −3), (2, −1) and (−1, 1)

Tamilnadu Board Math Term-IShow that the following points form a right–angled triangle.

(10, 0), (18, 0) and (10, 15)

Tamilnadu Board Math Term-IShow that the following points form a right–angled triangle.

(−11, 13), (−3, −1) and (4, 3)

Tamilnadu Board Math Term-IShow that the following points form an isosceles triangle.

(1, 3), (−3, –5) and (−3, 0)

Tamilnadu Board Math Term-IShow that the following points form an isosceles triangle.

(2, 3), (5, 7) and (1, 4)

Tamilnadu Board Math Term-IShow that the following points form an isosceles triangle.

(1, −2), (−5, 1) and (1, 4)

Tamilnadu Board Math Term-IFind the distance between the following pairs of points.

(0, 17) and (0, −1)

Tamilnadu Board Math Term-I