Three circles are described touching each other at two places externally. If the sides of triangle are $4 c m, 6 c m$ and $8 c m$ .Find the radii of the circles.

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Solution

Let the three circles have centers at $A, B$ and $C .$ The triangle formed by the centers be $A B C .$ Let the radii of the circles be $r_{1}, r_{2}$ and $r_{3}$ .
$A B = r_{1} + r_{2} = 4 c m$ (circles touch externally
$B C = r_{2} + r_{3} = 6 c m$
$C A = r_{3} + r_{1} = 8 c m$

Add all these :
$(A B + B C + C A) = 18 c m$
$2 (r_{1} + r_{2} + r_{3}) = 18 c m$
$r_{1} + r_{2} + r_{3} = 9 c m$

Subtracting each of the three equations from the above eq. we get:
$r_{3} = 5 c m$
$r_{1} = 3 c m$
$r_{2} = 1 c m$

Hence, this is the answer.

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Three circles are described touching each other at two places externally. If the sides of triangle are 4 cm, 6 cm and 8 cm.Find the radii of the circles.

Three circles are described touching each other at two places externally. If the sides of triangle are $4 c m, 6 c m$ and $8 c m$ .Find the radii of the circles.