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There is a rhombus of side 6 cm. One of the diagonal of the rhombus is equal to the side of the rhombus. An isosceles triangle is formed whose two equal sides are equal to another diagonal of the rhombus and the third side is equal to 8 cm. Find the area of the triangle.

There is a rhombus of side 6 cm. One of the diagonal of the rhombus is equal to the side of the rhombus. An isosceles triangle is formed whose two equal sides are equal to another diagonal of the rhombus and the third side is equal to 8 cm. Find the area of the triangle. Correct Answer 8√23 cm<sup>2</sup>

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Since one of the diagonal of the rhombus is equal to the side of the rhombus, it must be the smaller diagonal i.e. BD

Hence, triangle ADB is the equilateral triangle with angle = 60°

∴ ∠ADC = 180 - 60 = 120°

Using cos in rule in ΔADC;

cos 120° = (36 + 36 - AC2) / (2 × 6 × 6)

⇒ -1/2 = (72 - AC2)/72

⇒ -36 = 72 - AC2

⇒ AC = 6√3 cm

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ΔPQR is the isosceles triangle with equal side = 6√3 cm

∴ Height of triangle = √(108 - 16) = 2√23

∴ Area of the triangle = 1/2 × 8 × 2√23 = 8√23 cm2

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