Find the radius of the circle under these given conditions.

Question

I tried to solve the question by considering the circle to be an incircle of 2 triangles joined as in this manner :

Now, this is my solution :

Let AE =X then AF =x ( properties of tangent ) This means , EB = 5 -x Implies BC =5-x By doing this to the lower triangle, we get CD also as 5-x

BC+CD = BD = 6 Then , we find X. How do we calculate r after this? Have I gone wrong in my method ?

Answer 1

Alternative way:

Join the center of the circle $O$ to $E$. Then $$AO= AC-OC = 4-r$$ and $$OE=r,$$ so $$\sin \angle BAC = \frac 35 = \frac{OE}{AO} =\frac{r}{4-r} \implies r=\frac 32. $$ Note here I am considering the diagram you drew to refer to points.

Answer 2

Your intuition was right. Here is the formula and the correct solution:

$r=\frac{Area}{half-perimeter}=\frac{6\times 4:2}{(5+5+6):2}=\frac{12}{8}=\frac{3}{2}$