Question

AB is line segment of length 24 cm. C is its midpoint. On AB, AC and BC semicircles are described. Find the radius of the circle which touches all the three semicircles.

Answer 1

Let the required radius be r cm.

$O_1{O}_3$ = Radius of smaller semicircle + $r=\frac{24}{4}+r=6+r$

$O_{1}C$ = Radius of smaller semicircle = $\frac{24}{4}=6cm$

In right triangle $O_1{O}_3$ C,

$O_1{O}_3^2=O_1{C}^2+O_3{C}^{2}or{O}_3{C}^2=O_1{O}_3^2-O_1{C}^2=36+r^2+12r-36=r^2+12ror{O}_{3}C=(r^2+12r{)}^{\frac{1}{2}}Also,DC=D{O}_3+O_{3}Cor\frac{24}{2}=12=r+(r^2+12r{)}^{\frac{1}{2}}or12-r=(r^2+12r{)}^{\frac{1}{2}}$

Squaring both sides $144+r^2-24r=r^2+12r$
$or36r=144orr=\frac{144}{36}=4$

Hence, the radius of the circle which touches all three semicircles is 4 cm.