Final answer:
The length of HE in triangle TUV, where E, D, and H are midpoints, is determined using the midpoint theorem. As HD and EH are half the lengths of UV and TV, respectively, HE is established as 63 without necessitating complex equations, showcasing the simplicity of geometric principles.
Step-by-step explanation:
Given that points E, D, and H are the midpoints of the sides of triangle TUV, we can use the property that the line connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. Let HE = x. Since H and D are midpoints, HD = (1/2)UV = 49. Similarly, EH = (1/2)TV = (1/2)126 = 63. Therefore, HE = 63. The consistent application of this midpoint theorem simplifies the determination of HE. In summary, the strategic use of the midpoint theorem expedites the calculation of HE without delving into intricate equations, highlighting the elegance of geometric principles in solving problems related to triangles and their midpoints.