Question

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

Explanation:

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

By the Substitution Property of Equality, substitute \(x = 3\) into the expressions for \(TU\) and \(VW\).
For \(TU\): \(TU=3x\), so \(TU = 3(3)=9\).
For \(VW\): \(VW = 6 + x\), so \(VW=6+(3)=9\). Then \(TU = VW\).

Step2: Use the mid - point definition

Since \(U\) is the midpoint of \(\overline{TV}\), by the definition of a mid - point, \(TU=UV\).
Because \(TU = VW\) (from Step 1) and \(TU = UV\), by substitution, \(UV=VW\).

Step3: Conclude using 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 midpoint of \(\overline{UW}\).

Answer:

By the Substitution Property of Equality, substitute \(x = 3\) into \(TU = 3x\) and \(VW=6 + x\) to get \(TU=9\) and \(VW = 9\), so \(TU = VW\). Since \(U\) is the midpoint of \(\overline{TV}\), \(TU=UV\). By substitution (\(TU = VW\) and \(TU = UV\)), \(UV = VW\). Then, by the definition of the mid - point of a segment, \(V\) is the midpoint of \(\overline{UW}\).