The perpendicular from a vertex to the opposite side

Try this Drag the orange dots on each vertex
to reshape the triangle. Note the position of the altitude as you drag.

The altitude of a triangle is a line from a vertex to the opposite side, that is perpendicular to that side, as shown in the animation above. A triangle therefore has three possible altitudes. The altitude is the shortest distance from a vertex to its opposite side.

The word 'altitude' is used in two subtly different ways:

- It can refer to the line itself. For example, you may see "draw an altitude of the triangle ABC".
- As a measurement. You may see "the altitude of the triangle is 3 centimeters". In this sense it is used in way similar to the "height" of the triangle.

In most cases the altitude of the triangle is inside the triangle, like this:

Angles B, C are both acute |

Angle C is obtuse |

In the animation at the top of the page, drag the point A to the extreme left or right to see this.

It turns out that in any triangle, the three altitudes always intersect at a single point, which is called the orthocenter of the triangle. For more on this, see Orthocenter of a triangle.

The following two pages demonstrate how to construct the altitude of a triangle with compass and straightedge.

- Constructing the altitude of a triangle (altitude inside).
- Constructing the altitude of a triangle (altitude outside).

In the animation at the top of the page:

- Drag the point A and note the location of the altitude line. Drag it far to the left and right and notice how the altitude can lie outside the triangle.
- Drag B and C so that BC is roughly vertical. Drag A. Notice how the altitude can be in any orientation, not just vertical.
- Go to Constructing the altitude of a triangle and practice constructing the altitude of a triangle with compass and ruler.

- Triangle definition
- Hypotenuse
- Triangle interior angles
- Triangle exterior angles
- Triangle exterior angle theorem
- Pythagorean Theorem
- Proof of the Pythagorean Theorem
- Pythagorean triples
- Triangle circumcircle
- Triangle incircle
- Triangle medians
- Triangle altitudes
- Midsegment of a triangle
- Triangle inequality
- Side / angle relationship

- Perimeter of a triangle
- Area of a triangle
- Heron's formula
- Area of an equilateral triangle
- Area by the "side angle side" method
- Area of a triangle with fixed perimeter

- Right triangle
- Isosceles triangle
- Scalene triangle
- Equilateral triangle
- Equiangular triangle
- Obtuse triangle
- Acute triangle
- 3-4-5 triangle
- 30-60-90 triangle
- 45-45-90 triangle

- Incenter of a triangle
- Circumcenter of a triangle
- Centroid of a triangle
- Orthocenter of a triangle
- Euler line

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All rights reserved