Chapter 13.9, Problem 9ST2

To determine

To show: The quadrilateral $GOLD$ is a parallelogram.

Explanation of Solution

Given information: The vertices of a quadrilateral are $G\left(4,-1\right)$ , $O\left(0,0\right)$ , $L\left(2,6\right)$ , and $D\left(6,5\right)$ .

Proof:

The parallelogram is a quadrilateral with length of opposite sides of quadrilateral being equal.

The length of the side $GO$ can be calculated as:

  $\begin{array}{c}GO=\sqrt{{\left(4-0\right)}^2+{\left(-1-0\right)}^2}\\ =\sqrt{4^2+{\left(-1\right)}^2}\\ =\sqrt{16+1}\\ =\sqrt{17}\end{array}$

The length of the side $LD$ can be calculated as:

  $\begin{array}{c}LD=\sqrt{{\left(6-2\right)}^2+{\left(5-6\right)}^2}\\ =\sqrt{4^2+{\left(-1\right)}^2}\\ =\sqrt{16+1}\\ =\sqrt{17}\end{array}$

The length of the opposite sides $GO$ and $LD$ is equal.

The length of the side $GD$ can be calculated as:

  $\begin{array}{c}GD=\sqrt{{\left(6-4\right)}^2+{\left(5-\left(-1\right)\right)}^2}\\ =\sqrt{2^2+{\left(6\right)}^2}\\ =\sqrt{4+36}\\ =2\sqrt{10}\end{array}$

The length of the side $OL$ can be calculated as:

  $\begin{array}{c}OL=\sqrt{{\left(2-0\right)}^2+{\left(6-0\right)}^2}\\ =\sqrt{2^2+6^2}\\ =\sqrt{4+36}\\ =2\sqrt{10}\end{array}$

The length of the opposite sides $GD$ and $OL$ is equal. As the length of opposite side of the quadrilateral are of equal length, so it is a parallelogram.

Hence, the quadrilateral $GOLD$ is a parallelogram.