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 ( \begin{array} { l } { \text { substitution property of equality } } \\ { \text { select choice } } \\ { \text { midpoint theorem } } \\ { \text { symmetric property of equality } } \\ { \text { substitution property of equality } } end{array} ), ( tu = 3 x = 3 ( 3 ) = 9 ), and ( v w = 6 + x = 6 + ( 3 ) = 9 ), so ( \begin{array} { l } { \text { tu } = \text { vw } } end{array} ). also, because ( u ) is the midpoint of ( overline { t v } ), by the ( \begin{array} { l } { \text { midpoint theorem } } \\ { \text { symmetric property of equality } } \\ { \text { substitution property of equality } } end{array} ). thus, by substitution, ( u v = v w ). by the definition of ( \begin{array} { l } { \text { midpoint of a segment } } end{array} ), ( v ) is the midpoint of ( overline { u w } ).

Explanation:

Step1: Substitute \(x = 3\)

By the Substitution Property of Equality, \(TU=3x = 3\times3=9\) and \(VW = 6 + x=6 + 3=9\). So \(TU = VW\).

Step2: Use mid - point definition

Since \(U\) is the midpoint of \(\overline{TV}\), by the definition of midpoint \(TU=UV\).

Step3: Conclude \(UV = VW\)

Because \(TU = VW\) and \(TU = UV\), by substitution \(UV = VW\).

Step4: Prove \(V\) is mid - point

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

Answer:

\(V\) is the midpoint of \(\overline{UW}\) as shown in the above proof steps.