Chapter 10.3, Problem 11WE

To determine

why $GD=GE$ . Also, find the value of GA, and find the name of the three angles congruent to the $\angle GAB$ .

Answer to Problem 11WE

1. By multiplying $\frac{1}{3}$ on both sides of AD and BE, we get GD = GE.
2. GA = GB.
3. Three congruent angles to $\angle GAB$ are $\angle GED,\angle GDE,\angle GBA$ .

Explanation of Solution

Given:

  $\overline{AD}$ and $\overline{BE}$ are congruent medians of $\Delta ABC$ ,

Calculation:

1. According to the theorem,

GD = $\frac{1}{3}$ AD, and

GE = $\frac{1}{3}$ BE.

Given that $\overline{AD}$ and $\overline{BE}$ are congruent-medians of $\Delta ABC$ .

Hence, AD = BE.

Multiply both sides of equation with $\frac{1}{3}$ , one gets,

  $\frac{1}{3}$ AD = $\frac{1}{3}$ BE.

Hence, GD = GE.

2. According to the theorem,

GA = $\frac{2}{3}$ AD, and

GB = $\frac{2}{3}$ BE.

Given that $\overline{AD}$ and $\overline{BE}$ are congruent-medians of $\Delta ABC$ .

Hence, AD = BE.

Multiply both sides of equation with $\frac{2}{3}$ , one gets

  $\frac{2}{3}$ AD = $\frac{2}{3}$ BE.

Hence, GA = GB.

1. Consider the triangle GAB,
2. Since the two sides of the triangle GA and GB are equal.

  $\Delta GAB$ is an isosceles triangle.

Thus, $\angle GAB=\angle GBA$ .

Consider the triangle GED,

Since the two sides of the triangle are equal, that is, GE = GD. $\Delta GED$ is an isosceles triangle.

Thus $\angle GED$ = $\angle GDE$ .

From the figure, we have $\overline{ED}$ parallel to $\overline{AB}$ .

Hence, the alternate angles are also equal.

Thus, $\angle GAB$ = $\angle GDE$ .

Then, $\angle GAB$ = $\angle GED$ .

The congruent angles to $\angle GAB$ are $\angle GED,\angle GDE,\angle GBA$ .