Question

In semicircle of radius 18 cm , 2 semicircles are drawn of radius 9 cm with the same base and a circle is drawn in remaining space. find radius of circle

BFA and AFC are semi circles.
Find radius of circle with centre O.

Answer 1

Let $O_{3}x=x$ & medius of third circle = r
(It is assumed that $3^{rd}$ circle touches the 3 semi - circles. Otherwise there are numerous solutions).
$O_{3}x$ is a common tangent to small semi circles & $O_3$ lies as that.
$\therefore From\Delta O_3\times O_1:O_3{x}^2+O_1{x}^2=O_1{O}_3^2$
$x^2+9^2=(9+r{)}^2$
$or{x}^2-(9-r{)}^2=9^2$
$(x+9+r)(x-9-r)=81$
Now it can be seen that x + r = Radius of bigger semi - circle
= 18
$\therefore$ (18 + 9) (n - 9 - r) = 81
$x-r-9=\frac{88}{27}=3$
a - x - r = 12
& x + r = 18
(x + r) - (x - r) = 18 - 12 = 6
2r = 6
r = 3cm
$\therefore$ Radius of $3^{rd}$ circle = 3cm