A geometry problem by Toto Mub

Let AO = x, OC = x, OD = 1, DC = y. Then (x + 1)^2 + y^2 = 144 and y^2 + 1 = x^2. Substituting, x^2 + x - 72 = 0 = (x + 9)(x - 8), so x = 8. Ed Gray

So, we know that AC is 12.

Write then, OD as 1.

So AD is r+1, since AO is the radius.

We can up the Pythagorean theorm to prove that $(r+1)^2 + (DC)^2 = 144$

Then, we draw a line from O to C making line OC, which is also the radius.

We have another right triangle, and so we use the theorm again.

Giving $1+(DC)^2 = r^2$

We now know that $DC^2 = r^2 +1$

If we plug that in to the other triangle, then $2r^2 +2r = 144$

We can divide each side by 2, giving $r^2 + r = 72$

That makes $(r+9)(r-8) = 0$

So r = -9 or 8. But we can't have a negative radius, so the answer is 8.