The incircle of a triangle ABC touches AB at point P and has radius equal to 21.If AP = 23 and PB = 27, then find the perimeter of the triangle ABC.

Engineering

Mathematics

Area of a Triangle

Circumradius Inradius and exradius

Question

The incircle of a triangle ABC touches AB at point P and has radius equal to 21.If AP = 23 and PB = 27, then find the perimeter of the triangle ABC.

LiveFree JEE Main Previous Year Online Papers Live Free JEE Advance Previous Year Online Papers

College PredictorLive

Know your College Admission Chances Based on your Rank/Percentile, Category and Home State.

Get your JEE Main Personalised Report with Top Predicted Colleges in JoSA

r = 21

c = 50

$\tan \frac{B}{2} = \frac{21}{27} = \frac{7}{9}$

$\tan \frac{A}{2} = \frac{21}{23}$

$\tan \frac{C}{2} = \tan (90^{\circ} - \frac{A + B}{2}) = \cot (\frac{A}{2} + \frac{B}{2}) = \frac{1}{\tan (\frac{A}{2} + \frac{B}{2})}$

$\tan \frac{C}{2} = \frac{1 - \tan \frac{A}{2} \tan \frac{B}{2}}{\tan \frac{A}{2} + \tan \frac{B}{2}} = \frac{1 - \frac{21}{23} \cdot \frac{7}{9}}{\frac{21}{23} + \frac{7}{9}} = \frac{70}{350} = \frac{1}{5}$

using r = (s – c)tan $\frac{C}{2}$ ; 21 = (s – 50) · $\frac{1}{5}$

s – 50 = 105 ⇒ s = 155 ⇒ 2s = 345.