Question

In triangle $ABC,r=\frac{R}{6}$ and $r_1=7r$. Then the measure of angle $A=$

• A. $\frac{\pi }{12}$
• B. $\frac{\pi }{6}$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{3}$

Answer 1

Answer: (D)

$\frac{\pi }{3}$

According to the Formula related to Ex-radius of circles

$r_1-r=4Rsi{n}^2\frac{A}{2}$

Since, $r_1=7r$

$\Rightarrow 7r-r=4Rsi{n}^2\frac{A}{2}\Rightarrow 6r=4Rsi{n}^2\frac{A}{2}$

and since, $r=\frac{R}{6}$

$\Rightarrow 6\times \frac{R}{6}=4Rsi{n}^2\frac{A}{2}\Rightarrow \frac{1}{4}=si{n}^2\frac{A}{2}\Rightarrow sin\frac{A}{2}=\frac{1}{2}\Rightarrow \frac{A}{2}=si{n}^{-1}\frac{1}{2}={30}^o\Rightarrow A={60}^o$ or $\frac{\pi }{3}$

therefore, Answer is $D$