Chapter 10.1, Problem 31E

To determine

Prove that the diagonals of a parallelogram bisect each other.

Explanation of Solution

Given:

The diagonals of a parallelogram bisect each other.

Calculation:

Draw a parallelogram for the given statement.

  

Let the coordinates of the points $P$ (origin), $Q,R$ and $S$ are $\left(0,0\right)$ , $\left(m,n\right)$ , $\left(r+m,n\right)$ and $\left(r,0\right)$ respectively.

If the coordinates of $p_1$ and $p_2$ are $\left(p_1,q_1\right)$ and $\left(p_2,q_2\right)$ , respectively, then the midpoint of $\overline{p_1{p}_2}$ has coordinates $\left(\frac{p_1+p_2}{2},\frac{q_1+q_2}{2}\right)$ .

Coordinates of the midpoint $\left(PR\right)$

  $=\left(\frac{0+r+m}{2},\frac{0+n}{2}\right)=\left(\frac{r+m}{2},\frac{n}{2}\right)$

Coordinates of the midpoint $\left(SQ\right)$

  $=\left(\frac{r+m}{2},\frac{0+n}{2}\right)=\left(\frac{r+m}{2},\frac{n}{2}\right)$

Now, the coordinates of the segment $\overline{PR}=\overline{SQ}$ .

Hence thediagonals of a parallelogram bisect each other.