1. **Problem statement:** We have a circle with center $O$. Segments $UX$ and $UV$ are tangents to the circle, and $XV$ passes through the center $O$. Given $UV=12.6$ and $XV=12$, we need to find $XW$. 2. **Key properties and formulas:** - Tangents from a common external point to a circle are equal in length. So, $UX = UV$. - Since $XV$ passes through the center $O$, $XV$ is a diameter or a chord passing through the center. - The tangent at point $W$ touches the circle, so $XW$ is a tangent segment from $X$ to the circle. 3. **Understanding the figure:** - $UX$ and $UV$ are tangents from point $U$. - $XV$ passes through $O$, so $O$ lies on $XV$. - Points $U$, $X$, $W$, and $V$ are arranged such that $XW$ is a tangent segment from $X$. 4. **Using the tangent-secant theorem:** For a point outside a circle, the square of the length of the tangent segment equals the product of the lengths of the secant segment and its external part. Here, from point $X$, the tangent segment is $XW$, and the secant segment is $XV$ passing through $O$. So, $$XW^2 = XU \times XV$$ But we need to identify $XU$. 5. **Using the equality of tangents from $U$:** Since $UX$ and $UV$ are tangents from $U$, $$UX = UV = 12.6$$ 6. **Using the tangent-secant theorem at point $X$:** Since $XW$ is tangent and $XV$ is secant, $$XW^2 = XU \times XV$$ Substitute known values: $$XW^2 = 12.6 \times 12 = 151.2$$ 7. **Calculate $XW$:** $$XW = \sqrt{151.2} \approx 12.3$$ **Final answer:** $$XW \approx 12.3$$