Discussion

Explanation:

Consider the diagram below.

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In a rhombus diagonals perpendicularly bisect each other.

∴ Let OA = OC = x and OB = OD = y

In ∆AOB ⇒ x² + y² = 5²  …(1)

Also, area of the rhombus = ½ × 2x × 2y = 12

⇒ 2xy = 12   …(2)

(1) + (2)
⇒ x² + y² + 2xy = 25 + 12 = 37
⇒ (x + y)² = 37
⇒ x + y = √37   …(3)

(1) - (2)
⇒ x² + y² - 2xy = 25 - 12 = 13
⇒ (x - y)² = 13
⇒ x - y = √13   …(4)

Solving (3) and (4) we get,

2x = √37 + √13 and
2y = √37 - √13

∴ The longer diagonal = 2x = √37 + √13

Hence, option (b).

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