In $\triangle ABC$, the lengths of sides $AC$ and $AB$ are 12 cm and 5 cm, respectively. If the area of $\triangle ABC$ is $30 \mathrm{~cm}^{2}$ and $R$ and $r$ are respectively the radii of circumcircle and incircle of $\triangle ABC$, then the value of $2R+r$ (in cm) is equal to ______

JEE Main · 2021 · 16 Mar 2021 · Shift 2 · Properties of Triangle

△ A B C

, the lengths of sides

A C

and

A B

are 12 cm and 5 cm, respectively. If the area of

△ A B C

30 cm^{2}

and

R

and

r

are respectively the radii of circumcircle and incircle of

△ A B C

, then the value of

2 R + r

(in cm) is equal to ______

Answer: 15

From

Δ = \frac{1}{2} (A B) (A C) sin A

30 = \frac{1}{2} (5) (12) sin A \Rightarrow sin A = 1 \Rightarrow A = \frac{π}{2}

So the triangle is right angled at

A

and

B C = 5^{2} + 1 2^{2} = 13

cm. The hypotenuse is a diameter of the circumcircle, so

R = \frac{13}{2}

cm. With

s = \frac{5 + 12 + 13}{2} = 15

cm,

r = \frac{Δ}{s} = \frac{30}{15} = 2 cm

Hence

2 R + r = 13 + 2 = 15

cm.

JEE Main · 2021 · 16 Mar 2021 · Shift 2 · Integer answer