UPSC CDS 2 2021 Math Paper

Concept:The radii of three circles touching each other externally satisfy $AB = r_1 + r_3$, $BC = r_2 + r_3$, $CA = r_1 + r_2$.Explanation:Given $AB = 6\ \text{cm}$, $BC = 10\ \text{cm}$, $CA = 8\ \text{cm}$.Let the radii of circles with centres $A$, $B$, $C$ be $r_1$, $r_2$, $r_3$ respectively.When circles touch externally, the distance between centres equals the sum of their radii.Thus: $6 = r_1 + r_3$, $10 = r_2 + r_3$, $8 = r_1 + r_2$.Add all three equations: $(r_1 + r_3) + (r_2 + r_3) + (r_1 + r_2) = 6 + 10 + 8$.This simplifies to $2(r_1 + r_2 + r_3) = 24$.Dividing by 2 gives $r_1 + r_2 + r_3 = 12\ \text{cm}$.Therefore, the sum of the radii is $12$ cm.Answer:$12\ \text{cm}$ (Option C).