Answer:
To solve this problem, we can use the properties of tangents to a circle and similar triangles.
First, we can find the length of $YZ$ using the following steps:
1. Since $WX$ is tangent to the circumcircle at $X$, we know that $\angle WXY$ is a right angle (tangent and radius are perpendicular at the point of tangency).
2. We can use the property that if a line is parallel to one side of a triangle and intersects the other two sides, it creates similar triangles.
3. Let's denote the length of $WZ$ as $a$. Then, $XZ = 12 - a$.
4. Using the property of similar triangles, we have:
$\frac{XZ}{WX} = \frac{XY}{WZ}$
5. Substituting the given values, we get:
$\frac{12-a}{12} = \frac{15}{a}$
6. Solving for $a$, we get:
$a = 9$
7. Finally, we can find $YZ$ using the Pythagorean theorem:
$YZ^2 = XY^2 + ZW^2$
$YZ^2 = 15^2 + 9^2$
$YZ^2 = 225 + 81$
$YZ^2 = 306$
$YZ = \sqrt{306}$
$YZ \approx 17.49$
Therefore, the length of $YZ$ is approximately 17.49 units.
Step-by-step explanation:
