Altitude of a triangle
The perpendicular from a vertex to the opposite side
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.
It can be outside the triangle
In most cases the altitude of the triangle is inside the triangle, like this:
However, if one of the angles opposite the chosen vertex is
then it will lie outside the triangle, as below.
The angle ACB is opposite the chosen vertex A, and is
obtuse (greater than 90°).
|Angles B, C are both acute
The altitude meets the extended base BC of the triangle at right angles.
|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.
Things to try
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.
Other triangle topics
Perimeter / Area
Congruence and Similarity
Triangle quizzes and exercises
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