Circle O is tangent to AB at A, and angle ABD = 90 degrees. If AB = 12 and CD = 18, find the radius of the circle.
https://latex.artofproblemsolving.com/f/f/1/ff13ec3e917b2a46649871ef2354fc1290478bc0.png
+31 ok, no need to say it's due tmrw, cuz this is an aops thing
the solution:
Let's say that the radius is equal to r. Let's say the midpoint of CD is P. Then, we know that PC= 9. Draw OP, and notice that OP is perpendicular to CD. Also, since \(\angle ABD \) = 90, that means that AOPB is a rectangle, which implies AB=OP= 12. Using the pythagorean theorem, the radius is equal to \(\sqrt{9^2 + 12^2}, \text{which is equal to } \boxed{15}\). Just use pi r ^2 to find the area, which is 225 pi
