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Glencoe Geometry: Online Textbook Help13 chapters | 152 lessons

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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.

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.

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.

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*.

This is relevant to right triangles because the altitude of a right angle in a right triangle is the geometric mean of the two parts of the hypotenuse. In the picture, DB = the square root of (DC * DA) or (DB)^2 = DC * DA. How does that work? We've already shown that triangles CDB, BDA, and CBA are all similar; that means that their corresponding sides are proportional, so DB / DC = DA / DB = CB / BA.

If DB / DC = DA / DB, we can cross-multiply to get DB^2 = DC * DA, or DB = the square root of (DC * DA). You can use these relationships to find the lengths of missing parts in a triangle. For example, if you know the lengths of CD and DA, you can calculate the length of the altitude.

In this lesson, you learned about altitudes drawn from the right angle to the hypotenuse of a **right triangle**. The **altitude** divides the right triangle into two smaller triangles that are similar to each other and also similar to the big triangle. We proved this using the angle measures of the triangles. The altitude also represents the **geometric mean** between the two segments of the hypotenuse. We proved this using the proportional relationships between sides of the two **similar triangles**.

Terms | Definitions |
---|---|

Altitude | 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 |

Right triangle | a triangle with one right angle |

Similar triangles | have the same angle measures and proportional side lengths |

Geometric mean | The geometric mean of two numbers x and y is the square root of x * y |

Apply your knowledge of right triangle altitudes when it's time to:

- Recognize and find the altitude of a triangle
- Identify the similar angles when finding the altitude
- Solve for the geometric mean

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Glencoe Geometry: Online Textbook Help13 chapters | 152 lessons

- Geometric Mean: Definition and Formula 5:15
- Right Triangle Altitudes: Applications 5:02
- Special Right Triangles: Types and Properties 6:12
- Trigonometric Ratios and Similarity 6:49
- Practice Finding the Trigonometric Ratios 6:57
- Angles of Elevation & Depression: Practice Problems 4:41
- Law of Sines: Definition and Application 6:04
- Law of Cosines: Definition and Application 8:16
- List of the Basic Trig Identities 7:11
- Go to Glencoe Geometry Chapter 7: Right Triangles and Trigonometry

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