1. **Problem Statement:** We are given triangle $\triangle TUV$ with points $E$, $D$, and $H$ as midpoints of its sides. The side lengths are $UV=40$, $TV=50$, and the segment $HD=40$. We need to find the length of $HE$. 2. **Key Concept:** When points $E$, $D$, and $H$ are midpoints of the sides of a triangle, the segments connecting these midpoints form the *midsegment triangle*. Each midsegment is parallel to one side of the original triangle and its length is half the length of that side. 3. **Applying the Midsegment Theorem:** - Since $E$, $D$, and $H$ are midpoints, segment $HD$ is parallel to side $TU$ and $HD = \frac{1}{2} TU$. - Given $HD = 40$, we find $TU = 2 \times 40 = 80$. 4. **Finding $HE$:** - Segment $HE$ is parallel to side $UV$ and its length is half of $UV$. - Given $UV = 40$, then $HE = \frac{1}{2} \times 40 = 20$. 5. **Final Answer:** $$HE = 20$$ This uses the property that midsegments in a triangle are parallel to the third side and half its length, allowing us to find $HE$ directly from $UV$.