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Question

Let PQR be an isosceles triangles. Suppose that the sides PQ and PR are equal and let the length of PQ be K cm. If PQR=θ,cosθ=45 and area of triangle is M sq. cm, then which of the following is true about M?

A
M<K24
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B
M>K22
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C
K24<M<K22
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D
K22<M< K2
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Solution

The correct option is C K24<M<K22
Given that PQ= PR = K cm, PQS=θ,cosθ=45 and area of ΔPQR=M sq. cm.
Let PS be the perpendicular bisector on QR
PSQ=PSR=90 and QS=SR
Now in ΔPQS.
sin θ=PSPQ
1cos2θ=PSK (sin2θ+cos2θ=1)
PS=K1(45)2 (cos θ=45)
PS=K251625=K925
PS=3K5.......(i)
Now,cos θ=QSPQ
45=QSK
QS=4K5 ...(ii)
Therefore, area of ΔPQR=12×PS×QR
=12×PS×2QS (QR=2QS
=PS×QS
3K5×4K5 [From(i)and(ii)]
=12K225
M=1225K2
K24<M<K22
Hence, the correct answer is option (3).

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