Circle O is tangent to line AB at A, and angle ABD is 90 degrees. If AB = 12 and CD = 18, find the radius of the circle.
If you drew a line from O perpendicular to the line DC, it would intersect at the midpoint of CD.
Call that midpoint E.
The length of the radius is the length of OA, which is also the length of BE.
We know that CE is 9, half of CD.
We now need to find the length of BC.
There is a theorem that states that AB·BA = CB·BD.
If we call the length CB = x, then we have the equation: 12·12 = x(x + 18).
Solving this quadratic for x, we have that x = 6.
This means that BE = BC + CE = 6 + 9 = 15.