Question

In the given figure, a triangle PQR is drawn to circumscribe a circle of radius 6 cm such that the segments QT and TR into which QR is divided by the point of contact T are of lengths 12 cm and 9 cm respectively. If the area of △PQR = 189 cm² then the length of side PQ is [CBSE 2011]
(a) 17.5 cm
(b) 20 cm
(c) 22.5 cm
(d) 7.6 cm

Answer 1

We know that tangent segments to a circle from the same external point are congruent.
Therefore, we have
PS = PU = x
QT = QS = 12 cm
RT = RU = 9 cm
Now,

$$\begin{gathered} \mathrm{Ar}\left(△\mathrm{PQR}\right)=\mathrm{Ar}\left(△\mathrm{POR}\right)+\mathrm{Ar}\left(△\mathrm{QOR}\right)+\mathrm{Ar}\left(△\mathrm{POQ}\right) \\ \Rightarrow 189=\frac{1}{2}\times \mathrm{OU}\times \mathrm{PR}+\frac{1}{2}\times \mathrm{OT}\times \mathrm{QR}+\frac{1}{2}\times \mathrm{OS}\times \mathrm{PQ} \\ \Rightarrow 378=6\times (x+9)+6\times \left(21\right)+6\times \left(12+x\right) \\ \Rightarrow 63=x+9+21+x+12 \\ \Rightarrow 2x=21 \\ \Rightarrow x=10.5\mathrm{cm} \end{gathered}$$
Now, PQ = QS + SP = 12 + 10.5 = 22.5 cm
Hence, the correct answer is option (c ).