Question : In the given figure, ABC is a right-angled triangle, $∠$ ABC = 90° and $∠$ 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

Question

Question : Suppose $△ A B C$ be a right-angled triangle where $∠ A = 90 °$ and $A D ⊥ B C$ . If the area of $△ A B C = 40$ cm $^{2}$ and $△ A C D = 10$ cm $^{2}$ and $\overset{―}{A C} = 9$ cm, then the length of $B C$ is:
Option 1: 12 cm
Option 2: 18 cm
Option 3: 4 cm
Option 4: 6 cm

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Answer 1

Correct Answer: 6

Solution : Given: ABC is a right-angled triangle. $∠$ ABC = 90°, $∠$ ACB = 60°, and the radius of the smaller circle is 2 cm.

$∠$ ACB = 60° so, $∠$ ACP = $∠$ PCB = 30°
In $△$ CPD,
$\sin 30 ° = \frac{P D}{C P}$
⇒ CP = 2 × 2 = 4 cm
In $△$ CQE,
$\sin 30 ° = \frac{Q E}{C Q}$
⇒ 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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Question : In the given figure, ABC is a right-angled triangle, ∠ABC = 90° and ∠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: 4Option 2: 6Option 3: 4.5Option 4: 7.5

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Question : In the given figure, ABC is a right-angled triangle, $∠$ ABC = 90° and $∠$ 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