# 5.5: Altitudes

**At Grade**Created by: CK-12

**Practice**Altitudes

### Altitudes

An **altitude** is a line segment in a triangle from a vertex and perpendicular to the opposite side, it is also known as the height of a triangle. All of the red lines are examples of altitudes:

As you can see, an altitude can be a side of a triangle or outside of the triangle. When a triangle is a right triangle, the altitude, or height, is the leg. To construct an altitude, construct a perpendicular line through a point not on the given line. Think of the vertex as the point and the given line as the opposite side.

#### Investigation: Constructing an Altitude for an Obtuse Triangle

Tools Needed: pencil, paper, compass, ruler

- Draw an obtuse triangle. Label it
△ABC , like the picture to the right. Extend sideAC¯¯¯¯¯¯¯¯ , beyond pointA . - Construct a perpendicular line to
AC¯¯¯¯¯¯¯¯ , throughB .

The altitude does not have to extend past side

As was true with perpendicular bisectors, angle bisectors, and medians,the altitudes of a triangle are also concurrent. Unlike the other three, the point does not have any special properties.

**Orthocenter:** The point of concurrency for the altitudes of triangle.

Here is what the orthocenter looks like for the three triangles. It has three different locations, much like the perpendicular bisectors.

Acute Triangle |
Right Triangle |
Obtuse Triangle |
---|---|---|

The orthocenter is inside the triangle. | The legs of the triangle are two of the altitudes. The orthocenter is the vertex of the right angle. | The orthocenter is outside the triangle. |

#### Identifying Altitudes

Which line segment is an altitude of

In a right triangle, the altitude, or the height, is the leg. If we rotate the triangle so that the right angle is in the lower left corner, we see that leg

#### Determining Location of the Orthocenter

1. A triangle has angles that measure

Because all of the angle measures are less than

2. A triangle has an angle that measures

Because

### Examples

#### Example 1

True or false: The altitudes of an obtuse triangle are inside the triangle.

Every triangle has three altitudes. For an obtuse triangle, at least one of the altitudes will be outside of the triangle, as shown in the picture at the beginning of this section.

#### Example 2

Draw the altitude for the triangle shown.

The triangle is an acute triangle, so the altitude is inside the triangle as shown below so that it is perpendicular to the base.

#### Example 3

Draw the altitude for the triangle shown.

### Interactive Practice

### Review

Write a two-column proof.

- Given: Isosceles
△ABC with legsAB¯¯¯¯¯¯¯¯ andAC¯¯¯¯¯¯¯¯ BD¯¯¯¯¯¯¯¯⊥DC¯¯¯¯¯¯¯¯ andCE¯¯¯¯¯¯¯¯⊥BE¯¯¯¯¯¯¯¯ Prove:BD¯¯¯¯¯¯¯¯≅CE¯¯¯¯¯¯¯¯

For the following triangles, will the altitudes be inside the triangle, outside the triangle, or at the leg of the triangle?

△JKL is an equiangular triangle.△MNO is a triangle in which two the angles measure30∘ and60∘ .△PQR is an isosceles triangle in which two of the angles measure25∘ .△STU is an isosceles triangle in which two angles measures45∘ .

Given the following triangles, which line segment is the altitude?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 5.5.

### Notes/Highlights Having trouble? Report an issue.

Color | Highlighted Text | Notes | |
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### Image Attributions

Here you'll learn the definition of altitude and how to determine where a triangle's altitude will be found.

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