Question

Question

In triangle upper A upper C upper E, G is the centroid and BE = 9. Find BG and GE.

Triangle A C E is shown. Point B is on segment A C and a median connects points B and E. Point D is on segment C E and a median connects points D and A. Point F is on segment A E and a median connects points F and C. The medians intersect at the triangle's centroid, which is labeled as point G.

Answer 1

To find BG and GE, we can use the properties of triangles and centroids.

1) In a triangle, the medians divide each other in a 2:1 ratio. This means that BG:GD = 2:1 and GE:EF = 2:1.

2) Since BE is a median, BED is a 45-45-90 right triangle. Therefore, BD = DE.

3) Likewise, AE and CF are also medians, so AEF and CFE are 45-45-90 right triangles. Therefore, EF = 2 x GE and AE = 2 x GE.

4) Since BD = DE, triangle BED is an isosceles triangle, and BG is equal to half of BE. So, BG = 1/2 x BE = 1/2 x 9 = 4.5.

5) From the ratio in step 1, BG:GD = 2:1, and since BG = 4.5, GD = 4.5 x 1 = 4.5.

6) Since EF = 2 x GE, GE = 1/2 x EF. But in triangle AEF, AE = 2 x GE. So GE = 1/2 x EF = 1/2 x AE = 1/2 x 2 x GE = GE.

Therefore, BG = 4.5 and GE = 4.5.