In tringle ACE,G is the centroid and BE = 15. BE is a median. Find BG and GE

Answer 1

In triangle ACE, the centroid G divides each median into two parts, where the distance from the centroid to the vertex is twice that from the centroid to the midpoint.

Let BG = x and GE = 2x.

Since BE is a median, it divides AC into two congruent segments, making AE = CE.

Now, since G is the centroid, AG = 2(GE) and CG = 2(BG).

So, AG = 2(2x) = 4x and CG = 2(x) = 2x.

Since AE = CE, AE + CE = AC.

4x + 2x = 6x = AC.

Since BE = 15, we can write:

AC = 2(BE) = 2(15) = 30.

Hence, 6x = 30.

Dividing both sides of the equation by 6 gives:

x = 30/6 = 5.

Therefore, BG = x = 5 and GE = 2x = 2(5) = 10.

So, BG = 5 and GE = 10.