Question

points e, d, and h are the midpoints of the sides of \\( \triangle tuv \\). \\( uv = 108 \\), \\( tv = 132 \\), and \\( hd = 108 \\). find tu. \\( tu = \square \\)

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that the length of a mid - segment of a triangle is half the length of the side it is parallel to. In \(\triangle TUV\), since \(H\) and \(D\) are mid - points, \(HD\) is a mid - segment. But we can also note that if \(E\), \(D\), and \(H\) are mid - points, we can use another approach.
Let's recall that the perimeter - related or side - length relationships in a triangle with mid - points. The key formula (derived from the mid - segment properties) for the side \(TU\): If we consider the fact that the side \(TU\) can be found using the double of the non - mid - segment related value. Given \(HD = 108\) (but actually, using the full - length relationship). Since \(E\), \(D\), \(H\) are mid - points, we know that \(TU=2\times HD\) (by the mid - segment theorem extended for the full side length when considering the mid - point connections in the triangle).

Step2: Calculate \(TU\)

Substitute \(HD = 108\) into the formula \(TU = 2\times HD\). So \(TU=2\times108\).

$$TU = 216$$

Answer:

216