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Geometry: High School15 chapters | 160 lessons

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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.

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.

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.

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.

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.

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

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.

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.

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*.

These triangles have the same area. Why? Because the median hits *BC* in its exact middle, dividing it into two equal parts. So the areas of the newly created triangles are equal.

Now let's add back medians *BE* and *CD*. These triangles all look different. But they all have the same area. *ADG* = *AGE* = *EGC* = *CGF* = *BGF* = *BGD*. No one has a bigger triangle piece than anyone else.

Just like with our centroid, our triangle areas remain in harmony no matter how we modify our triangle. As long as the lines from each vertex hit the midpoint of the side opposite them, forming medians, we'll always create equal triangles.

In summary, we learned that a median is a line segment drawn from the vertex, or corner, of a triangle to the midpoint of the opposite side.

Each triangle has three medians. They always meet at a single point, which we call a centroid. The centroid is the center of mass of the triangle.

Finally, we looked at the area of our triangles. A single median gives us two triangles with equal areas. When we have three medians, we get six triangles; all of these triangles have the same area.

Six kids, six triangles, let's enjoy this moment.

As this video lesson concludes, you could be able to:

- Illustrate a median line segment
- Remember the number of medians in a triangle
- Find the centroid of a triangle

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Geometry: High School15 chapters | 160 lessons

- Area of Triangles and Rectangles 5:43
- Perimeter of Triangles and Rectangles 8:54
- How to Identify Similar Triangles 7:23
- Angles and Triangles: Practice Problems 7:43
- Triangles: Definition and Properties 4:30
- Classifying Triangles by Angles and Sides 5:44
- Interior and Exterior Angles of Triangles: Definition & Examples 5:25
- Constructing the Median of a Triangle 4:47
- Constructing Triangles: Types of Geometric Construction 5:59
- Properties of Concurrent Lines in a Triangle 6:17
- Go to High School Geometry: Properties of Triangles

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