Question

If X is the centroid of △ABC, GE=12, BF=57, GC=18, and BC=32. Find GD, CD, BG, GF, AG, and BE.

A) GD = 6, CD = 24, BG = 19.5, GF = 6, AG = 24, BE = 38
B) GD = 6, CD = 12, BG = 38, GF = 12, AG = 38, BE = 19.5
C) GD = 12, CD = 6, BG = 38, GF = 6, AG = 38, BE = 19.5
D) GD = 12, CD = 24, BG = 19.5, GF = 12, AG = 19.5, BE = 38

Answer 1

To find the lengths GD, CD, BG, GF, AG, and BE in a triangle with X as the centroid and given lengths, we can use the properties of a centroid and ratios of medians.

Step-by-step explanation:

To find the lengths GD, CD, BG, GF, AG, and BE, we first need to understand the properties of a centroid in a triangle. The centroid is the point of concurrency of the medians of a triangle, and it divides each median into two parts in the ratio 2:1.

Let's apply this knowledge to the given triangle ABC. BC is the median, and X is the centroid. Therefore, GD:DX = 2:1. Since DX = 6 (half of 12), GD = 2 * 6 = 12.

Similarly, we can find that CD = 2 * 18 = 36 and BG = 2 * 57 = 114. The remaining lengths can be found using the concept of similar triangles and the fact that the centroid divides each median into ratios of 2:1.