Q11. - Geometry

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  $\angle GAB$

Step-by-Step Solution

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 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 $\angle GAB$  are  $\angle GED,\angle GDE,\angle GBA$.