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Median, Altitude, and Angle Bisectors of a Triangle

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  • 0:06 Triangle Fashion
  • 0:50 Median
  • 1:39 Altitude
  • 3:22 Angle Bisector
  • 4:20 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.

Line segments in triangles are just clutter; they have special names and qualities depending on how they're drawn. In this lesson, we'll learn about medians, altitudes and angle bisectors.

Triangle Fashion

I don't claim to have much fashion expertise. Well, not when it comes to people. I don't know a clutch from a tote or why you'd need so many purses.

But when it comes to triangles, I'm much more confident. Your basic triangle looks like this.

Basic Triangle

It has three sides and three angles. It has corners that we call vertices, or a vertex for just one.

This triangle looks just fine. But it's lacking... something. In this lesson, we're going to learn about triangle fashion. We're going to add lines to our triangles that take them from 'meh' to 'wow.' These lines serve different roles in triangles, just like purses. Well, I think like purses. I'm still not sure about those.

Median

First, let's draw a line from a vertex across a triangle like this.

Median (triangle)

We call this a median. A median is a line segment drawn from the vertex of a triangle to the midpoint of the opposite side.

It splits the opposite side into two equal line segments. We know it's a median if we have those equal line segments. It's like knowing someone paid attention while getting dressed if his socks match.

Depending on how formal our triangle's outing is, we can have up to three medians, one from each vertex. When we draw three medians, they always meet at a single point.

Medians are kind of like belts. They divide triangles in two. In fact, the two new triangles formed by adding a median have equal areas. And these six triangles formed by three medians also have equal areas.

Altitude

So if medians are belts, what about earrings? Ok, this metaphor is venturing further into the unknown for me, but what happens with big earrings that are, um, dangly? They hang straight down, perpendicular to the ground, right? And why do they do that? Gravity.

And what's the triangle equivalent? Altitudes. An altitude is a perpendicular line segment drawn from a vertex of a triangle to the opposite side.

Altitude (triangle)

In our triangle here, if we draw a line from A perpendicular to the opposite side, it's an altitude. We could do this from any vertex, but we most commonly see it from the top.

Think of it like that earring. Gravity pulls it straight down. Unlike a median, an altitude doesn't necessarily split the opposite side into equal segments. In fact, it only will in two types of triangles.

In an equilateral triangle, all the angles are equal. Here, the altitude comes right down the middle and, in fact, is the same as the median.

For an equilateral triangle, the median cuts the side in half and is the same as an altitude.
equilateral triangle with median

This is also true in an isosceles triangle. Well, it's true for one of our altitudes. If we draw the other two, they clearly aren't also medians.

Only one of the possible medians in an isoceles triangle is also an altitude.
Isoceles triangle with three possible medians

In a right triangle, two of the altitudes are actually sides of the triangle, since the sides already meet at right angles.

A right triangle has two altitudes that are also sides.
Right triangle with sides as two altitudes

Here, we don't have matching socks, do we?

Also, an altitude isn't always inside a triangle. When there's an obtuse angle, we have an altitude that's actually outside the triangle. This is like an earring of someone leaning to the side for some reason. And instead of matching socks, it's like wearing a blue sock on one foot and hat on the other.

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