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Constructing the Median of a Triangle

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  • 0:06 Dividing Equally
  • 0:50 Medians
  • 1:55 Centroid
  • 3:02 Equal Triangles
  • 4:10 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

If you have one triangle and want to divide it, you can use a median line. Medians have special geometric properties that we'll learn about in this lesson.

Dividing Equally

Meet Maddie and Emma. They're twins. Everything Maddie gets, Emma needs to get, too, and vice versa. When Maddie got a bike, so did Emma. When Emma started playing soccer, so did Maddie. When Maddie got braces, well, Emma was okay letting Maddie have that one.

Maddie and Emma's parents are used to dividing everything equally. That's one of those skills that are also useful in geometry. Consider this triangle that Maddie has.

 Triangle with vertices~

It's so great. It has an A, B and C on its corners. Each corner even has a fancy name: vertex (or vertices for more than one).

Emma wants one too. But that's the last triangle. Oh no! We're going to need to divide it equally.

Medians

Let's draw a line from A to the middle of the opposite side, BC. We call this a median and can define it as a line drawn from the vertex of a triangle to the midpoint of the opposite side.

A line from a vertex to the midpoint of the opposite side is a median
triangle with median

When we do this, what happens to BC? Since this new point, F, is the midpoint, we've made two equal line segments. BF = FC. The midpoint is just the point in the middle. Makes sense, right?

And now we have two triangles: one for Maddie and one for Emma. Peace is restored.

But wait, now four of their friends show up. You know how kids are with their triangles. Everyone wants one. Fortunately, triangles like things in threes - three sides, three angles, three vertices and, yes, three medians.

Let's add a median from B to AC and call it BE. Then let's add a median from C to AB and call it CD. And look what we get.

Drawing three median lines creates six triangles
A triangle with three median lines drawn creates six triangles

Now we have six triangles. They're smaller, sure, but we again have momentary peace.

Centroid

While everyone is content with their triangles, let's look at what happened. Our three medians, AF, BE and CD all hit the midpoints of the sides of the triangle. That's what makes them medians. So in addition to BF equaling FC, AD = DB and AE = EC.

But what else? Notice that they all meet at one point inside the circle, G. It doesn't matter what our triangle looks like. In any triangle, the three medians meet at one point.

The medians of a triangle cross at one point, the centroid
triangle medians cross at the centroid

We call this point the centroid. This is officially defined as the center of mass of a two-dimensional polygon.

The word centroid reminds me of android. Imagine a triangular, two-dimensional android. Hmm, not much good for blowing up Death Stars or saving Sigourney Weaver from aliens. But Centroid the android has one cool trick.

If you draw three medians on him, they meet at one point: Centroid's centroid.

If we take Centroid and balance him on a stick, with the stick right on the centroid, he'll balance perfectly. That's what the center of mass is.

Equal Triangles

Ok, I think we've used up our good will with Emma, Maddie, and their friends. They're arguing now over their six triangle pieces. They think they all got different sizes.

But this is the cool thing about medians. Let's go back to our original triangle. Remember how we drew line AF, our first median? That gave us two triangles, ABF and ACF.

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