Altitude of a triangle - Math Open Reference (original) (raw)

The perpendicular from a vertex to the opposite side

Try this Drag the orange dots on each vertexto 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:

Angles B, C are both acute

However, if one of the angles opposite the chosen vertex is obtuse, then it will lie outside the triangle, as below. The angle ACB is opposite the chosen vertex A, and is obtuse (greater than 90°).

Angle C is obtuse

The altitude meets the extended base BC of the triangle at right angles.

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

Orthocenter

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.

Constructions

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).

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

General

• 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 / Area

• 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

Triangle types

• 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

Triangle centers

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

Congruence and Similarity

• Congruent triangles

Solving triangles

• Solving the Triangle
• Law of sines
• Law of cosines

Triangle quizzes and exercises

• Triangle type quiz
• Ball Box problem
• How Many Triangles?
• Satellite Orbits

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