Chapter 7.5, Problem 2MRE

a.

To determine

To complete: The given statement.

Answer to Problem 2MRE

S is the midpoint of $\overline{RT}$ and $\overline{SU}\parallel \overline{RV}$

Explanation of Solution

Given information: W and U are the midpoint of $\overline{RV}$ and $\overline{VT}$ , $\overline{SW}\parallel \overline{VT}$

  

According to converse of midpoint theorem,

If a line is drawn through the midpoint of a side of a triangle parallel to the second side, then it will bisect the third side.

Now in $\Delta RTW$ , W is a midpoint of $\overline{RV}$ and $\overline{SW}\parallel \overline{VT}$

So,

S is a midpoint of $\overline{RT}$

According to midpoint theorem,

The line joining the midpoints of two side of a triangle is parallel to the third side.

As S is a midpoint of $\overline{RT}$ and U is a midpoint of $\overline{VT}$ ,

  $\overline{SU}\parallel \overline{RV}$

b.

To determine

To complete: The given statement.

Answer to Problem 2MRE

  $x=2$

Explanation of Solution

  

According to Midpoint Theorem,

The line joining the midpoints of two side of a triangle is parallel to the third side and equal to half of it.

As S is a midpoint of $\overline{RT}$ and W is a midpoint of $\overline{RV}$ ,

  $\overline{SW}=\frac{1}{2}\overline{TV}$

Put $\overline{SW}=3x+8,\overline{TV}=12x+4$

So,

  $\begin{array}{l}3x+8=\frac{1}{2}\left(12x+4\right)\\ \Rightarrow 6x+16=12x+4\\ \Rightarrow 16-4=12x-6x\\ \Rightarrow 12=6x\\ \Rightarrow x=2\end{array}$

c.

To determine

To complete: The given statement.

Answer to Problem 2MRE

  $y=4$

Explanation of Solution

  

According to Midpoint Theorem,

The line joining the midpoints of two side of a triangle is parallel to the third side and equal to half of it.

As S is a midpoint of $\overline{RT}$ and U is a midpoint of $\overline{TV}$ ,

  $\overline{SU}=\frac{1}{2}\overline{RV}=\overline{RW}$

Put $\overline{SU}=9y-19,\overline{RW}=4y+1$

So,

  $\begin{array}{l}9y-19=4y+1\\ \Rightarrow 9y-4y=1+19\\ \Rightarrow 5y=20\\ \Rightarrow y=4\end{array}$