1. **Problem Statement:** Given a circle centered at $O$, with tangent segments $HK$ and $HI$ touching the circle at points $J$ and $I$ respectively. The segment $KI$ passes through the center $O$. Given $HK=19.4$ and $KI=13$, find the length $KJ$. 2. **Key Concepts and Formulas:** - Tangent segments from a common external point are equal in length. Thus, $HJ = HI$ and $KJ = KI$ if $J$ and $I$ are tangent points. - Since $KI$ passes through the center $O$, $O$ lies on $KI$. - The segment $HK$ is composed of $HJ + JK$. 3. **Step-by-step Solution:** - Since $HK$ and $HI$ are tangent segments from points $H$ and $I$ respectively, and $J$ and $I$ are tangent points, the tangent lengths from $H$ to the circle are equal: $HJ = HI$. - Given $HK = 19.4$, and $HK = HJ + JK$, we want to find $KJ$. - The segment $KI$ passes through $O$, and $KI = 13$. - Because $J$ lies on the tangent line $HK$ between $H$ and $K$, and $HI$ is tangent at $I$, the tangent lengths from $K$ to the circle are equal: $KJ = KI$. - Therefore, $KJ = 13$. 4. **Final Answer:** $$KJ = 13$$