Chapter 7, Problem 22CCSR

To determine

To find: the value of $x$ .

Answer to Problem 22CCSR

The value of $x$ is 4.

Explanation of Solution

Given information:

A parallelogram has vertices $\left(-3,2\right),\left(0,7\right),\left(7,7\right)$ and $\left(x,2\right)$ .

Calculation:

Now, construct a parallelogram using the given coordinates in the question as given below.

  

To find the value of $x$ , use fact that the diagonals of a parallelogram bisect each other.

That is, the midpoint O will be same for both the diagonals of the given parallelogram, the calculation is done by using following formula:

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

So, the midpoint of both the diagonal will be equal:

  $\begin{array}{c}\text{midpointof}\text{AC}=\text{midpoint}\text{of}\text{DB}\\ \left(\frac{-3+7}{2},\frac{2+7}{2}\right)=\left(\frac{x+0}{2},\frac{2+7}{2}\right)\\ \left(\frac{4}{2},\frac{9}{2}\right)=\left(\frac{x}{2},\frac{9}{2}\right)\\ \left(2,4,5\right)=\left(\frac{x}{2},4.5\right)\end{array}$

Now, compare the coordinate of $x$ :

  $\begin{array}{c}\frac{x}{2}=2\\ x=4\end{array}$

Hence, the value of $x$ is 4.