- #1

The Head

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The Problem is #16 in the attached picture. Essentially, I need to find the length of BC using information about congruency and the location of the centroid. I've been able to show a whole bunch of things, but nothing that gets me close to actually finding out the missing side length.

I began by drawing all of the other medians through point G, which forms triangles AGE and CGE. Some of the things I logically inferred were (note, when I use "=" below I sometimes mean congruent-- I do know the difference):

BG=GD=2GE=2ED

AE=EC

triangle AED = triangle CEG

triangle AEG = triangle CED

AG=6 (because of congruent triangles) and the distance along the median from G to the side of BC is 3 (because distance from centroid to side length is 1/3 length of median)

These congruent triangles a lot of sides and angles to be congruent, but I can't really make any progress on specific values. Please help me make some real progress with this problem!

I began by drawing all of the other medians through point G, which forms triangles AGE and CGE. Some of the things I logically inferred were (note, when I use "=" below I sometimes mean congruent-- I do know the difference):

BG=GD=2GE=2ED

AE=EC

triangle AED = triangle CEG

triangle AEG = triangle CED

AG=6 (because of congruent triangles) and the distance along the median from G to the side of BC is 3 (because distance from centroid to side length is 1/3 length of median)

These congruent triangles a lot of sides and angles to be congruent, but I can't really make any progress on specific values. Please help me make some real progress with this problem!