Question

proof in the figure at the right, u is the midpoint of \\( \overline{tv} \\), and \\( x = 3 \\).
write a paragraph proof to prove that v is the midpoint of \\( \overline{uw} \\).

because u is the midpoint of \\( \overline{tv} \\), we know that \\( tu= \\)
substituting \\( x = 3 \\), we find \\( tu= \\) and \\( uv= \\)
next, we can find the lengths of uv and vw by substituting \\( x = 3 \\) into the expressions given in the
diagram.
so \\( uv= \\) and \\( vw= \\)
because \\( uv = vw \\), point v divides \\( \overline{uw} \\) into two congruent segments.
therefore, v is the midpoint of \\( \overline{uw} \\).

Explanation:

Step1: Use mid - point property

Since \(U\) is the mid - point of \(\overline{TV}\), by the definition of a mid - point, \(TU = UV\).

Step2: Calculate \(TU\) and \(UV\)

Given \(TU = 3x\) and \(x = 3\), then \(TU=3\times3 = 9\). Since \(TU = UV\), \(UV = 9\).

Step3: Calculate \(VW\)

Given \(VW=6 + x\) and \(x = 3\), then \(VW=6 + 3=9\).

Answer:

Because \(U\) is the midpoint of \(\overline{TV}\), we know that \(TU = UV\). Substituting \(x = 3\), we find \(TU=9\) and \(UV = 9\). Next, we can find the lengths of \(UV\) and \(VW\) by substituting \(x = 3\) into the expressions given in the diagram. So \(UV = 9\) and \(VW = 9\). Because \(UV = VW\), point \(V\) divides \(\overline{UW}\) into two congruent segments. Therefore, \(V\) is the midpoint of \(\overline{UW}\).