UPSC CDS 1 2021 Math Paper

let the sides AB, AC and BC of a triangle ABC are 4 cm, 6 cm and 8 cm. let the touching pointing of the circles are L, M and N such that AN = AL, BL = BM and CN = CM.
Let AN = a cm then CN = CM = 6 – a cm
Then BM = BL = 8 – (6 – a) = 2 + a
And AL = AN = 4 – ( 2 + a) = 2 - a
But AN = a then
a = 2 – a
a = 1 cm
sum of the radii of the circle = AN + BM + CL = 1 + 5 + 3 = 9 cm