# Geometry: Finding a Side Length in Triangle Using Centroid

• The Head
In summary, the problem is to find the length of BC using information about congruency and the location of the centroid. The individual has drawn all the medians through point G, forming triangles AGE and CGE, and has inferred that BG=GD=2GE=2ED, AE=EC, triangle AED = triangle CEG, triangle AEG = triangle CED, and AG=6. However, they are unable to make progress on finding specific values. They then ask for help and it is mentioned that the position of the centroid in a right triangle is b/3 and h/3 away from the right angle. This information allows them to use the Pythagorean Theorem to find the missing length of BC.
The Head
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!

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I moved the thread to our homework section.
The Head said:
AG=6
That is a good start.

What do you know about the position of the centroid in a right triangle?

Ahh I can see now! It's b/3 and h/3 away from the right angle and then we can just use the Pythagorean Theorem. Thank you.

## 1. What is a centroid in a triangle?

A centroid is a point of concurrency in a triangle that is located at the intersection of the three medians. A median is a line segment that connects a vertex to the midpoint of the opposite side.

## 2. How do you find the centroid of a triangle?

To find the centroid of a triangle, you need to first find the coordinates of the three vertices. Then, you can use the formula (x1 + x2 + x3)/3 and (y1 + y2 + y3)/3 to find the coordinates of the centroid.

## 3. What is the formula for finding a side length using the centroid?

The formula for finding a side length using the centroid is 2/3 of the length of the median that intersects that side. This can be written as s = (2/3)*m, where s is the side length and m is the length of the median.

## 4. Can the centroid be outside of the triangle?

No, the centroid will always be located within the triangle as long as the triangle is not degenerate (meaning it has three collinear points). This is because the centroid is the intersection of the medians, which are always inside the triangle.

## 5. Are there any other methods for finding a side length using the centroid?

Yes, another method is to use the distance formula to find the length of the median, and then multiply it by 3/2 to find the side length. This can be written as s = 3/2 * √[(x2-x1)^2 + (y2-y1)^2], where x1, x2, y1, and y2 are the coordinates of the two endpoints of the median.

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