of 
is a right angle. The sides of 
, and the arc of the semicircle on 
. What is the radius of the semicircle on 
?![[asy] import graph; pair A,B,C; A=(0,8); B=(0,0); C=(15,0); draw((0,8)..(-4,4)..(0,0)--(0,8)); draw((0,0)..(7.5,-7.5)..(15,0)--(0,0)); real theta = aTan(8/15); draw(arc((15/2,4),17/2,-theta,180-theta)); draw((0,8)--(15,0)); dot(A); dot(B); dot(C); label("$A$", A, NW); label("$B$", B, SW); label("$C$", C, SE);[/asy]](https://latex.artofproblemsolving.com/2/6/2/2624acb19756f1b697f71413e9840070908905da.png)
Thomas A. answered • 11/24/23
Tutor
5.0
(128)
Math and science made easy - learn from a engineering student
Solution 1
If the semicircle on 
were a full circle, the area would be 
.
, therefore the diameter of the first circle is 
.
The arc of the largest semicircle is 
, so if it were a full circle, the circumference would be 
. So the 
.
By the Pythagorean theorem, the other side has length 
, so the radius is 
Solution 2
We go as in Solution 1, finding the diameter of the circle on 
and . Then, an extended version of the theorem says that the sum of the semicircles on the left is equal to the biggest one, so the area of the largest is 
, and the middle one is 
, so the radius is 
.
