Question : In the given figure, ABC is a right-angled triangle, $\angle$ABC = 90° and $\angle$ACB = 60°. If the radius of the smaller circle is 2 cm, then what is the radius (in cm) of the larger circle?
Option 1: 4
Option 2: 6
Option 3: 4.5
Option 4: 7.5
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Correct Answer: 6
Solution :
Given: ABC is a right-angled triangle. $\angle$ABC = 90°, $\angle$ACB = 60°, and the radius of the smaller circle is 2 cm.
$\angle$ACB = 60° so, $\angle$ACP = $\angle$PCB = 30°
In $\triangle$CPD,
$\sin30°=\frac{PD}{CP}$
⇒ CP = 2 × 2 = 4 cm
In $\triangle$CQE,
$\sin30°=\frac{QE}{CQ}$
⇒ CQ = 2QE
Now, CQ = CP + OP + OQ
⇒ 2QE = 4 + 2 + QE
⇒ OQ = 6 cm
So, the radius of the larger circle is 6 cm.
Hence, the correct answer is 6.
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