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The area of a rhombus is 2016 square centimetres, and its side is 65 cm. What are the lengths of the diagonals (in cm) respectively?

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125, 35

126, 32

132, 26

135, 25

answer is B.

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Detailed Solution

Let the length of the diagonals be

d 1

and

d 2

.
Given that the area of the rhombus is

2016 c m 2

. Then,

A = 12 × d 1 × d 2 2016 = 12 × d 1 × d 2

4032 = d 1 × d 2

… (1)
By using the property, we know that the diagonal of rhombus bisects each other at

90 ∘

. Then

(65) 2 = d 1 2 2 + d 2 2 2 4225 = d 12 + d 22 4

d 12 + d 22 = 16900

… (2)
We know that,

(d 1 + d 2) 2 = d 12 + d 22 + 2 d 1 d 2

(d 1 + d 2) 2 = d 12 + d 22 + 2 d 1 d 2 = 16900 + 2 (4032) = 16900 + 8064 d 1 + d 2 = 24964 d 1 + d 2 = 158 c m

Similarly, we know that,

(d 1 − d 2) 2 = d 12 + d 22 − 2 d 1 d 2

(d 1 − d 2) 2 = d 12 + d 22 − 2 d 1 d 2 = 16900 − 2 (4032) = 16900 − 8064 d 1 − d 2 = 8836 d 1 − d 2 = 94 c m

From the above equation, we have

d 1 + d 2 = 158 d 1 = 158 − d 2

… (3)

d 1 − d 2 = 94 c m (158 − d 2) − d 2 = 94 − 2 d 2 = 94 − 158 d 2 = 642 d 2 = 32 c m

Substitute the above value in the equation (3), and we have

d 1 = 158 − d 2 = 158 − 32 = 126 c m

Hence, the correct option is 2.

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