Final answer:
The question requires finding the length of segment HE given midpoints on a triangle and lengths of its sides. Usually, a midsegment is half the length of the side it parallels. However, the given information suggests a typo since HD should not equal UV if it is a midsegment.
Step-by-step explanation:
The problem involves finding the length of segment HE in triangle TUV, given that points E, D, and H are midpoints of the sides of the triangle and given the lengths UV, TV, and HD. Using the properties of midsegments in a triangle, we know that a segment connecting two midpoints of a triangle is parallel to the third side and is half its length.
Since UV = 78 and HD is a midsegment connecting midpoints of TU and TV, HD must be half of UV, which is already given as 78. This suggests a possible typo in the question since HD and UV should not be equal if HD is indeed a midsegment.
However, if we proceed with the assumption that HD and UV are equal due to a unique situation where TUV might be an isosceles triangle with TV = TU, allowing HD to equal UV, we still can't directly find HE without additional information.
The segment HE is the midsegment corresponding to side TV, which means that normally HE would be half the length of TV. But without confirmation of the relationship between HD and UV and the specific type of triangle TUV, we cannot conclude the length of HE.