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Question

ABCD is a rectangle such that ∠CAB=60∘ and AC=a units. The area of rectangle ABCD is ______.

A
32a2
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B
23a2
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C
34a2
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D
38a2
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Solution

The correct option is C √34a2ABCD is a rectangle. Hence∠ABC=900. So, in ΔABC, right-angled at B, Sin θ=oppositehypotenuseSin 600=BCAC On substituiting the value of Sin 600=√32 and AC = a units, we get ⇒√32=BCa⇒BC=√32a---------(i) Cos θ=adjacenthypotenuseCos 60∘=ABAC On substituiting the value of Cos 600=12 and AC = a units, we get ⇒12=ABa⇒AB=a2---------(ii) From(i) and (ii), So, area of the rectangle = length × breadth =AB×BC=a2×√32a=√3a24

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