Chapter 12.8, Problem 18E

To determine

To Prove: The value $\overline{DG}\cong \overline{EH}$ .

Explanation of Solution

Given:

The given diagram is shown in Figure 1

  

Figure 1

The value of $\Delta DEA$ H is the midpoint of $\overline{DA}$ , G is the midpoint of $\overline{EA}$ .

Calculation:

Consider the midpoint of the line segment $\overline{DA}$ is,

  $\begin{array}{c}H=\left(\frac{-2h+0}{2},\frac{0+2k}{2}\right)\end{array}$

    $\begin{array}{c}=\left(-h,k\right)\end{array}$

Consider the midpoint of the line segment $\overline{EA}$ is,

  $\begin{array}{c}G=\left(\frac{2h+0}{2},\frac{0+2k}{2}\right)\end{array}$

    $\begin{array}{c}=\left(h,k\right)\end{array}$

The length of the side DG is,

  $\begin{array}{c}DG=\sqrt{{\left(h-2k\right)}^2+{\left(k-0\right)}^2}\end{array}$

    $\begin{array}{c}=\sqrt{9h^2+k^2}\end{array}$

The length of the side EH is,

  $\begin{array}{c}EH=\sqrt{{\left(2h+h\right)}^2+{\left(0-k\right)}^2}\end{array}$

    $\begin{array}{c}=\sqrt{9h^2+k^2}\end{array}$

Hence, proved that $\overline{DG}\cong \overline{EH}$ .