Radius of a Semi Circle -AMC 8, 2017 - Problem 22

Here O is the center of the semi circle. Join o and D(where D is the point where the circle is tangent to the triangle ) and Join OB.

Can you now finish the problem ..........

Now the $\triangle ODB $and $\triangle OCB$ are congruent

can you finish the problem........

Let x be the radius of the semi circle

Now the $\triangle ODB$ and $\triangle OCB$ we have

OD=OC

OB=OB

$\angle ODB$=$\angle OCB$= 90 degree`

so $\triangle ODB$ and $\triangle OCB$ are congruent (by RHS)

BD=BC=5

And also $\triangle ODA$ and $\triangle BCA$ are similar....

$\frac{8}{12}$=$\frac{x}{5}$

i.e x =$\frac{10}{3}$