Chapter 11.2, Problem 6QR

To determine

The missing coordinate so that four point from a parallelogram $ABCD$ .

Answer to Problem 6QR

The required missing coordinate is $C=\left(8,6\right)$ .

Explanation of Solution

Given information:

Given points

  $\begin{array}{l}A=\left(1,1\right)\\ B=\left(3,5\right)\\ C=\left(8,b\right)\\ D=\left(6,2\right)\end{array}$

Formula used:

Slop of a line connecting two point $P\left(a,b\right)$ and $Q\left(c,d\right)$ :

  $PQ=\frac{c-a}{d-b}$

Calculation:

It is known that, in parallelogram slop of the opposite sides are equal.

Thus, in the given parallelogram $ABCD$ , the slop of the line segment $AB$ and $CD$ must be equal.

Now, calculate the slop of $AB$ , use the formula $PQ=\frac{c-a}{d-b}$ .

  $\begin{array}{c}AB=\frac{5-1}{3-1}\\ =\frac{4}{2}\\ =2\end{array}$

Similarly, calculate $CD$ .

  $\begin{array}{c}CD=\frac{2-b}{6-8}\\ =\frac{b-2}{2}\end{array}$

Since $AB=CD$ , we get,

  $\begin{array}{c}\frac{b-2}{2}=2\\ b-2=4\\ b=6\end{array}$

Conclusion:

Hence, the required coordinate is $C=\left(8,6\right)$ .