In a city, there is a circular park. There are four points of entry into the park, namely - P, Q, R and S. Three paths were constructed which connected the points PQ, RS, and PS. The length of the path PQ is 10 units, and the length of the path RS is 7 units. Later, the municipal corporation extended the paths PQ and RS past Q and R respectively, and they meet at a point T on the main road outside the park. The path from Q to T measures 8 units, and it was found that the angle PTS is 60 . Find the area (in square units) enclosed by the paths PT, TS, and PS.
Explanation:
The following diagram represents the given scenario.
Let, TR = x
∴ 8 × 18 = x (7 + x)
∴ x2 + 7x – 144 = 0
∴ x = 9 or x = – 16
But x cannot be negative.
Hence, x = 9
A(∆TPS) = 12 × 18 × 16 × sin 60°
= 72√3 sq.units
Hence, option (c).
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