Chapter 1.1, Problem 57E

To determine

To prove: diagonals of the given parallelogram intersect at their midpoints.

Explanation of Solution

Given:

The parallelogram:

Concept Used:

Midpoint Formula: Midpoint of the line segment joining the two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is given by:

  $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$

To prove that diagonals of the given parallelogram intersect at their midpoints, it is enough to show that both diagonals have same midpoints.

First find the midpoint of the diagonal whose end points are $\left(0,0\right)$ and $\left(a+b,c\right)$ .

So applying Midpoint formula on the line segment joining the two points $\left(0,0\right)$ and $\left(a+b,c\right)$ .

  $\begin{array}{c}\text{Midpoint}=\left(\frac{0+\left(a+b\right)}{2},\frac{0+c}{2}\right)\\ =\left(\frac{a+b}{2},\frac{c}{2}\right)\end{array}$

Now find the midpoint of the diagonal whose end points are $\left(a,0\right)$ and $\left(b,c\right)$ .

So applying Midpoint formula on the line segment joining the two points $\left(a,0\right)$ and $\left(b,c\right)$ .

  $\begin{array}{c}\text{Midpoint}=\left(\frac{a+b}{2},\frac{0+c}{2}\right)\\ =\left(\frac{a+b}{2},\frac{c}{2}\right)\end{array}$

Conclusion:

Both the diagonals have same midpoints, so the diagonals must intersect at their midpoints.