Question

In the following figure, PQ is the diameter of the circle with centre O and QR is the tangent to the circle at Q.
PR intersects the circle at S and the tangent to the circle as S intersects QR at T. If the diameter of the circle is 20 cm and TR=6 cm, find the approximate length of the line segment PR.

• A. 33.2 cm
• B. 26.16 cm
• C. 24.57 cm
• D. 22.5 cm

Answer 1

The correct option is D

22.5 cm

Clearly. $\angle$PSQ=90${}^{\circ }$ [Angle in a semi circle]
Also, TQ=TS [Tangents from an external point are equal ] ... (1)
Let $\angle$TQS=$\angle$TSQ=x
Clearly, $\angle$TSR=90${}^{\circ }$ - x
Also in $\Delta$QSR, $\angle$QRS=90${}^{\circ }$-x
:.TR=TS ... (2)
From (1) and (2), QT=TR
Given that TR=6 cm.
$\Rightarrow$ QR=QR+TR =6+6=12cm
Now, the length of the line segment PR = $\sqrt{P{Q}^2+Q{R}^2}$
=$\sqrt{{20}^2+{12}^2}=\sqrt{544}=\sqrt{16\times 34}$
=4$\sqrt{34}$ cm=22.5 cm