In the given figure ABC is right angle triangle, right angled at B. A semi -circle is drawn inside the triangle

In the given figure ABC is right angle triangle, right angled at B. A semi -circle is drawn inside the triangle. Find the radius of semicircle if AO = 4cm & OC = 5cm?

(a) 15/√41

(b)  20/√41

(c)  25/√41

(d) 30/√41

 Sol:

∆ADO ~ ∆OEC

∴ in ∆ADO, sinθ = r/4

and in ∆OFC, cosθ = r/5

Now, sin²θ + cos²θ = 1

=>r2/16+r2/25=1

=>r2=16*25/41

=>r=4*5/√41=20/√41