Right Triangle Altitudes: Applications

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  • 0:01 Altitude Review
  • 0:56 Similar Traingles
  • 2:51 Geometric Mean
  • 4:28 Lesson Summary
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Lesson Transcript
Instructor: Elizabeth Foster

Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.

In this lesson, you'll learn about some special properties that govern the altitude of the right angle in a right triangle. There's also a refresher on what all those terms mean, if you need one.

Altitude Review

If you've ever been in an airplane, you've experienced one kind of altitude. But an altitude is also a technical term for a part of a triangle. The altitude of a triangle is a line drawn from one angle of the triangle all the way through the triangle, so that it makes a right angle with the opposite side and divides the big triangle into two smaller triangles. Every triangle has three altitudes, one starting from each corner.

But in this lesson, we're going to talk about some qualities specific to the altitude drawn from the right angle of a right triangle. A right triangle is a triangle with one right angle. A right angle is a 90-degree angle, often indicated by a little box in the corner. The altitudes of right angles have some special qualities, so fasten your seat belt securely around your hips, and let's take a look.

Similar Triangles

First of all, altitudes separate the big triangle into two smaller triangles. A special property of right angle altitudes is that all three of these triangles are similar triangles, which have the same angle measures and proportional side lengths. The proof for this is a little involved, so just to make things easier, we'll label this triangle with points A through D, and angles E through J.

Start off with the definition of an altitude: it intersects the side at a right angle. So both angle G and angle H must be 90 degrees. It might be tempting to say that J and E are equal because they look kind of the same, but an altitude doesn't necessarily split the angle into two equal parts; they could be different. So don't go by just the way they look in the picture! We know they add up to 90 degrees, but we don't know that they're the same.

So what do we know? We know that all triangles add up to 180 degrees. Since angle G accounts for 90 of those degrees, E + F = 90. Now look at the big triangle; in the big triangle, the angles are I, J + E, and F. We know that J + E = 90, so I + F must account for the other 90 degrees in the triangle. Now we have:

E + F = 90

E = 90 - F

I + F = 90

I = 90 - F

Since E and I are both equal to 90 - F, angles E and I must be equal to each other. Now we have two equal angles, so we know that F must be equal to J, and the triangles must be similar. The two smaller triangles are similar to each other, and also to the big triangle.

Geometric Mean

Another property of the altitude of a right angle in a right triangle has to do with the geometric mean. The geometric mean of two numbers x and y is the square root of x * y. If m represents the geometric mean of x and y, we can say either m = the square root of x * y or m^2 = x * y.

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