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Q.

The area of the right angled triangle in terms of its circum radius and in radius is

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a

$R (r + 2 R)$

b

$R (2 r + R)$

c

$r (r + 2 R)$

d

$r (2 r + R)$

answer is A.

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Detailed Solution

$\begin{array}{l} \frac{c}{2} = R, \frac{a + b - c}{2} = r \\ ∴ \frac{a + b - c}{2} = r ∴ \frac{a + b - c}{2} = r \\ \Rightarrow a + b = 2 r + 2 R \\ \Rightarrow a^{2} + b^{2} + 2 a b = 4 r^{2} + 4 R^{2} + 8 r R \end{array}$

$\begin{array}{l} \Rightarrow c^{2} + 2 a b = 4 r^{2} + 4 R^{2} + 8 r R \\ \Rightarrow \frac{1}{4} 2 a b = r^{2} + 2 R r \\ \Rightarrow \frac{1}{2} a b = r (r + 2 R) \end{array}$

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