We are given a circle with center $A$ and radius $x$. A line segment $\overline{BC}$ is tangent to the circle at point $B$. We are also given that $BC = 24$ and $CD = 18$. We need to find the value of $x$, which is the radius of the circle.

1. Problem Description

We are given a circle with center $A$ and radius $x$. A line segment $\overline{BC}$ is tangent to the circle at point $B$. We are also given that $BC = 24$ and $CD = 18$. We need to find the value of $x$, which is the radius of the circle.

2. Solution Steps

Since $\overline{BC}$ is tangent to the circle at point $B$, the radius $\overline{AB}$ is perpendicular to $\overline{BC}$. Therefore, $\triangle ABC$ is a right triangle with a right angle at $B$. The length of $AC$ is $AD + DC = x + 18$. We can apply the Pythagorean theorem to $\triangle ABC$:

$AB^2 + BC^2 = AC^2$

Substitute the given values:

$x^2 + 24^2 = (x + 18)^2$

$x^2 + 576 = x^2 + 36x + 324$

$576 = 36x + 324$

$36x = 576 - 324$

$36x = 252$

$x = \frac{252}{36}$

$x = 7$

3. Final Answer

The value of $x$ is 7.