Question

These exploratory exercises can be done using a computer with a program that draws and measures geometric figures.

2. a. Draw any $▵ABC.$Draw the three medians. They should intersect in one point, as shown in the diagram at the right. Find the ratios $\frac{AG}{AD},\frac{BG}{BE},\frac{CG}{CF}.$ What do you notice?

Short answer

The final answer is:

$\begin{array}{l}AG:AD::2:3\\ BG:BE::2:3\\ CG:CF::2:3\end{array}$

Step-by-step solution

Step 1. Given information:

$AD,BE$ and $CE$ are median of triangle $ABC$

Step 2. Concept used:

We use basic geometric and angle concept.

Step 3. Applying the concept:

The median of the triangle divides the other median in the ratio of 2:1

So,

$\begin{array}{l}\frac{AG}{AD}\\ \frac{AG}{AG+GD}\\ AG=2x\\ GD=x\\ \frac{2x}{2x+x}\\ \frac{2x}{3x}\\ \frac{2}{3}\\ AG:AD::2:3\end{array}$

Similarly,

$\begin{array}{l}BG:BC::2:3\\ CG:CF::2:3\end{array}$