1. **Problem statement:** Given a circle centered at $O$, with tangent segments $VY$ and $VW$ from point $V$ to the circle, and segment $YW$ passing through $O$. Given $VY=8.7$ and $YW=6$, find the length $YX$ where $X$ lies on segment $VW$. 2. **Key properties and formulas:** - Tangents from a common external point to a circle are equal in length, so $VY = VW = 8.7$. - Since $YW$ passes through the center $O$, $Y$ and $W$ are endpoints of a diameter, so $YW$ is the diameter of the circle. - Point $X$ lies on $VW$ between $V$ and $W$. 3. **Find the radius $r$ of the circle:** Since $YW$ is the diameter, $$r = \frac{YW}{2} = \frac{6}{2} = 3.$$ 4. **Use the right triangle formed by the tangent:** The tangent length $VY$ relates to the distance $VO$ (distance from $V$ to center $O$) and radius $r$ by the Pythagorean theorem: $$VY^2 = VO^2 - r^2.$$ Substitute known values: $$8.7^2 = VO^2 - 3^2,$$ $$75.69 = VO^2 - 9,$$ $$VO^2 = 75.69 + 9 = 84.69,$$ $$VO = \sqrt{84.69} = 9.2.$$ 5. **Find $VX$ and $XW$ on segment $VW$:** Since $VW = VY = 8.7$ and $X$ lies on $VW$, and $YW$ passes through $O$, $X$ is the foot of the perpendicular from $V$ to $YW$ (because $VY$ and $VW$ are tangents and $X$ lies between $V$ and $W$). 6. **Calculate $YX$:** Since $Y$ and $W$ lie on the diameter, and $X$ lies on $VW$, the length $YX$ is the difference between the radius and the projection of $VO$ onto $YW$. Because $VO = 9.2$ and radius $r=3$, the length $YX$ is: $$YX = VO - r = 9.2 - 3 = 6.2.$$ **Final answer:** $$\boxed{6.2}$$