Q40. - Geometry

Question

For the following figure, $\mathit{E}$ is the midpoint of $\overline{\mathbf{D}\mathbf{F}}$. Find the value $\mathit{z}$, where $\mathit{G}\mathit{E}\mathbf{=}\mathit{z}$, $\mathit{E}\mathit{H}\mathbf{=}\mathbf{2}\mathit{z}\mathbf{-}\mathbf{4}$ and $\mathit{G}\mathit{H}\mathbf{=}\mathit{z}\mathbf{+}\mathbf{6}$ . 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 5, $\mathit{G}\mathit{E}\mathbf{=}\mathbf{5}$ and $\mathit{E}\mathit{H}\mathbf{=}\mathbf{6}$, also $\mathit{E}$ is not 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:

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

Step 4- Simplify the equation.

Simplify the equation as follows:

$\begin{array}{c}z+6=3z-4\\ z-3z=-4-6\\ -2z=-10\end{array}$

Divide the equation both sides by $-2$.

$\begin{array}{c}-2z=-10\\ \frac{-2z}{-2}=\frac{-10}{-2}\\ z=5\end{array}$

Therefore, the value of $z$ is 5.

Step 5- Step description.

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

$\begin{array}{l}GE=z\\ GE=5\end{array}$

$\begin{array}{c}EH=2z-4\\ EH=2\left(5\right)-4\\ EH=10-4\\ EH=6\end{array}$

Since the value of $GE$ and $EH$ are not the same, therefore $E$ is not the midpoint of $\overline{GH}$.