Question

In the figure, $\overline{AD}$ and $\overline{BE}$ are congruent medians of $▵ABC$ .

1. Explain why $GD=GE.$
2. $GA=?$
3. Name three angles congruent to $\angle GAB$

Short answer

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

Step-by-step solution

Step 1. Given information:

$\overline{AD}$ and $\overline{BE}$ are congruent medians of $▵ABC$.

Step 2. Concept used.

Basic concept of geometry related to triangles.

Step 3. applying the concept.

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.

1. 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.

c.Consider the triangle GAB,

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 role="math" $\angle GAB$are $\angle GED,\angle GDE,\angle GBA$.