JEE Main 2021 (Online) 16th March Evening ShiftQuestion 1
In $\triangle ABC$, the lengths of sides $AC$ and $AB$ are $12$ cm and $5$ cm, respectively. If the area of $\triangle ABC$ is $30$ 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 $\_\_\_\_\_\_$ .
Trigonometric Ratio & Identities
Answer$15$
Solution
Concept. The area shows $\sin A=1$, so the triangle is right-angled at $A$. Formulas used. (1) Area $=\frac12bc\sin A$; right triangle: $R=\frac{\text{hypotenuse}}{2}$, $r=\frac{\Delta}{s}$Step 1 - Angle $A$. $\frac12\cdot12\cdot5\sin A=30$ gives $\sin A=1$. So $BC=\sqrt{144+25}=13$. Step 2 - Radii. $R=\frac{13}{2}$, $s=15$, $r=\frac{30}{15}=2$. Step 3 - Value. $2R+r=13+2=15$. Answer: 15