A Part of a semicircle between the two legs of a right angle triangle In a right angled triangle, a semicircle is drawn such that its diameter lies on the hypotenuse and its center divides the hypotenuse into two segments of lengths 15 and 20.Find the length of the arc of the semicircle between the points at which the legs touch the semicircle.

![A Part of SemiCirc](

Let the tangent length shown in sketch be T. The power of circle

$$ T^2 = (15-R) (15+R) \tag{1}$$

From similar triangles, (radius/hypotenuse) of right side right angled triangle:

$$ \frac{T}{R}= \frac{15}{20}= \frac{3}{4} \tag{2}$$

Solving

$$ R= 12, \; T = 9 \tag{3} $$

Arc Length is quarter circle $$ =\pi \, R/ 2 = \frac{ \pi \cdot 12}{2} = 6 \pi \tag{4}$$