Question

proof in the figure, u is the midpoint of tv, and x = 3. write a paragraph proof to prove that v is the midpoint of uw.
given: u is the midpoint of tv, and x = 3.
prove: v is the midpoint of uw.
proof: by the midpoint theorem, tu = 3x = 3(3) = 9, and vw = 6 + x = 6 + (3) = 9, so tv = uv. also, because u is the midpoint of tv, by the definition of congruence, tv = uw. thus, by substitution, uv = vw. by the definition of midpoint of a segment, v is the midpoint of uw.

Explanation:

Step1: Calculate the lengths of \(TU\) and \(VW\)

Given \(x = 3\), for \(TU\), we use the formula \(TU=3x\). Substituting \(x = 3\), we get \(TU = 3\times3=9\). For \(VW\), we use the formula \(VW=6 + x\). Substituting \(x = 3\), we get \(VW=6+3 = 9\).

Step2: Use the mid - point property

Since \(U\) is the mid - point of \(TV\), by the definition of a mid - point, \(TU=UV\). We found \(TU = 9\), so \(UV = 9\).

Step3: Compare \(UV\) and \(VW\)

We have \(UV = 9\) and \(VW=9\). So, \(UV = VW\) by substitution.

Step4: Apply the mid - point definition

By the definition of the mid - point of a segment (a point that divides a segment into two congruent segments), since \(UV = VW\), \(V\) is the mid - point of \(UW\).

Answer:

By substituting \(x = 3\) into the expressions for \(TU\) (\(TU = 3x\)) and \(VW\) (\(VW=6 + x\)), we find \(TU=9\) and \(VW = 9\). Because \(U\) is the mid - point of \(TV\), \(TU = UV\) (by the definition of a mid - point). So \(UV=9\). Since \(UV = 9\) and \(VW = 9\), \(UV = VW\) (by substitution). By the definition of the mid - point of a segment (a point that divides a segment into two equal parts), \(V\) is the mid - point of \(UW\).