1. **Problem Statement:** We are given triangle $ABC$ with segment $DE$ inside it. Points $D$ and $E$ lie on sides $AB$ and $AC$ respectively. Segment $DE$ is parallel to side $BC$ and is a midsegment of the triangle. We need to find the value of $x$, the length of segment $DE$, given that $BC = 26$. 2. **Relevant Formula and Rule:** The Midsegment Theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and its length is half the length of that third side. Mathematically, if $DE$ is a midsegment parallel to $BC$, then: $$ DE = \frac{1}{2} BC $$ 3. **Applying the Formula:** Given $BC = 26$, substitute into the formula: $$ DE = \frac{1}{2} \times 26 = 13 $$ 4. **Conclusion:** The length of segment $DE$ is $13$. Therefore, the value of $x$ is 13, which corresponds to option B.