Let G denote the centroid of triangle ABC. If triangle ABG is equilateral with side length 2, then determine the perimeter of triangle ABC.

I drew the diagram, but it doesn't really help....

+Also Why would AoPS ban someone from taking their classes? They make like 400 bucks/person/class.

You know that the sides of ABG are all 2. Then, since the centroid is the point of intersection of the medians, median CD which intersects line AB would create two equal lengths; AD=BD=1.

Then, because it is equilateral, <GBA=<GAB=<BGA=60 degrees, so the median would split <BGA in half, so <BGD=30 degrees. Because the sum of a triangle's interior angles is 180 degrees, triangle BGD is a right triangle, so length GD=sqrt(3).

Furthermore, you know that the centroid divides up medians into a ratio of 2:1, so CG=2sqrt(3).

I think you can figure the rest out.

@ The first person Anonymous, if you recognize the question from AoPs, doesn't that make you cheating too???

@AoPS Only AoPS impersonators "ban" people from taking AoPS. If a mod/teacher doesn't want you posting on other websites, they would simply tell you to use the message board.

To determine the perimeter of triangle ABC, we can approach the problem using the properties of centroids in triangles.

The centroid of a triangle divides each median in a ratio of 2:1. This means that if triangle ABG is equilateral with side length 2, the centroid G divides side AB into two segments AG and GB, with AG = 2/3 of AB and GB = 1/3 of AB.

Since triangle ABG is equilateral with side length 2, we can conclude that AG = GB = 2/3 * 2 = 4/3.

Therefore, if AG = 4/3, GB = 4/3, and AB = 2, we can express the length of side AC as AC = AG + GB = 4/3 + 4/3 = 8/3.

Similarly, we can find the length of side BC as BC = BG + GC. Since GB = 4/3, we need to determine the length of GC.

By the property of centroids, GC divides side BC into two segments GB and BC in a ratio of 2:1. Since GB = 4/3, we can express BC as 3/2 * GB = 3/2 * 4/3 = 2.

Therefore, BC = 2.

Now that we know the lengths of sides AC and BC, we can find the perimeter of triangle ABC as the sum of these two sides along with side AB:

Perimeter of triangle ABC = AB + AC + BC
= 2 + 8/3 + 2
= 16/3 + 6/3 + 6/3
= 28/3

Hence, the perimeter of triangle ABC is 28/3.

what a chilly cow

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i love this site! great alternate to the AoPS message board! <3

Are you by any chance enrolled (or were enrolled) in AoPS Intro to Geometry course? Because I recognize all of the problems you post...

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