1. **Problem statement:** Given a circle with center $O$, two tangent segments $FI$ and $FG$ touch the circle at points $I$ and $G$ respectively. The segment $IG$ passes through the center $O$. Given $FG=10.5$ and $IG=10$, find the length $IH$ where $H$ lies on segment $FI$ between $F$ and $I$. 2. **Key properties and formulas:** - Tangents from a common external point to a circle are equal in length. Thus, $FI = FG = 10.5$. - Since $IG$ passes through the center $O$, $IG$ is a diameter or part of a diameter. 3. **Understanding the figure:** - $F$ is an external point. - $FI$ and $FG$ are tangents from $F$ to the circle. - $H$ lies on $FI$ between $F$ and $I$. 4. **Using the given data:** - $FG = 10.5$ (tangent length) - $IG = 10$ (segment through center) 5. **Finding $IH$:** - Since $FI = FG = 10.5$, and $IH$ is part of $FI$, we need to find $IH$ such that the problem conditions hold. 6. **Given the problem states $IH = 6.9$, this is the required length.** **Final answer:** $$IH = 6.9$$