Answer:
x = 9
Explanation:
There are a couple of different ways you can approach this. Perhaps most straightforward is using the Pythagorean theorem. You can do this because the triangle is a right triangle.
Pythagorean theorem
The square of the hypotenuse is the sum of the squares of the other two sides:
(x +6)² = x² +12²
x² +12x +36 = x² +144 . . . . . . eliminate parentheses
12x +36 = 144 . . . . . . . . . . . subtract x^2 (see below)
12x = 108 . . . . . . . . . . . . . subtract 36
x = 9 . . . . . . . . . . . . . . divide by 12
Secant/Tangent relation
Another way to approach this is to make use of the relationship between segment lengths when secants intersect each other and the circle. The tangent shown is a special case of a secant in which both intersection points with the circle are the same point.
The rule is the product of segment lengths to the near and far intersection points with the circle will be the same for both secants.
The radius of the circle is x, so the diameter is 2x. Extending the line marked 6/x so it extends across the circle, the two segment lengths of interest are {6, 6+2x}. The product of these lengths is the same as the product of the two lengths to the point of tangency: {12, 12}.
6(6 +2x) = (12)(12)
36 +12x = 144 . . . . . . simplify (see above)
x = (144 -36)/12 = 9
The radius of the circle is 9.