Q39. - Geometry

Question

For the following figure, $\mathit{E}$ is the midpoint of   $\overline{\mathbf{D}\mathbf{F}}$. Find the value  $\mathit{z}$, where  $\mathit{G}\mathit{E}\mathbf{=}\mathit{z}\mathbf{+}\mathbf{2}$, $\mathit{E}\mathit{H}\mathbf{=}\mathbf{2}\mathit{z}\mathbf{-}\mathbf{6}$ and $\mathit{G}\mathit{H}\mathbf{=}\mathbf{20}$. Find the $\mathit{G}\mathit{E}$, $\mathit{E}\mathit{H}$ and state whether $\mathit{E}$ is the midpoint of  $\overline{\mathbf{G}\mathbf{H}}$.

Step-by-Step Solution

Answer

The value of $z$ is 8, $\mathit{G}\mathit{E}\mathbf{=}\mathbf{10}$ and $\mathit{E}\mathit{H}\mathbf{=}\mathbf{10}$ also $\mathit{E}$ is the midpoint of  $\overline{\mathbf{G}\mathbf{H}}$

Step 1- Apply the concept of the length of the line.

The total length of the line is the sum of the length of all parts of a line.

Step 2- Check the figure.

Consider the figure.

Here the length of $GH$ is the sum of $GE$ and $EH$

Step 3- Find the value of z .

Since  $GH=GE+EH$

Equate the values as follows:

$\begin{array}{l}20=z+2+2z-6\\ 20=3z-4\end{array}$ 

Step 4- Simplify the equation.

Simplify the equation as follows: 

$\begin{array}{c}20=3z-4\\ 3z=20+4\\ 3z=24\end{array}$

Divide the equation both sides by 3.

$\begin{array}{c}3z=24\\ \frac{3z}{3}=\frac{24}{3}\\ z=8\end{array}$

Therefore, the value of $z$ is 8.

Step 5- Step description.

Find the values of $GE$ and $EH$ as follows:

$\begin{array}{l}GE=z+2\\ GE=8+2\\ GE=10\end{array}$

$\begin{array}{c}EH=2z-6\\ EH=2\left(8\right)-6\\ EH=16-6\\ EH=10\end{array}$

 

Since the value of $GE$ and $EH$ are same, therefore $E$ is the midpoint of $\overline{GH}$.