Chapter 5.3, Problem 23WE

To determine

To prove: $PX=QY$

Explanation of Solution

Given information: In $\Delta ABC$ , midpoints of $\overline{AB},\overline{AC},\overline{BC}$ are $X,Y,Z$ respectively. P is the midpoint of $\overline{BZ}$ and Q is the midpoint of $\overline{CZ}$

Proof:

  

In $\Delta ABC$ ,

midpoints of $\overline{AB},\overline{AC}$ are $X,Y$ respectively

According to midpoint theorem,

In a triangle, the line segment joining the midpoints of two sides of the triangle is parallel to the third side and is also half of the length of the third side.

So,

  $XY\parallel PQ(i)$

Also,

  $XY=\frac{1}{2}BC(ii)$

As P is the midpoint of $\overline{BZ}$ , $BP=PZ=\frac{1}{2}BZ$

As Q is the midpoint of $\overline{CZ}$ , $ZQ=QC=\frac{1}{2}ZC$

As midpoint of $\overline{BC}$ is $Z$ , $BZ=ZC$

  $\begin{array}{l}\Rightarrow \frac{1}{2}BZ=\frac{1}{2}ZC\\ \Rightarrow PZ=ZQ\end{array}$

Therefore,

  $BP=PZ=ZQ=QC$

  $\begin{array}{l}PZ+ZQ=BP+QC\\ \Rightarrow PQ=\frac{1}{2}BC(iii)\end{array}$

From $(ii)$ and $(iii)$ ,

  $XY=PQ(iv)$

From $(i),(iv)$ ,

  $XY=PQ,XY\parallel PQ$

In a quadrilateral, if one pair of sides is equal and parallel then it is a parallelogram.

Therefore,

  $XYQP$ is a parallelogram.

So,

  $PX=QY$

(Opposite sides of a parallelogram are equal)