how to find the altitude of a triangle

The altitude of a triangle, or height, is a line from a vertex to the opposite side, that is perpendicular to that side. It can also be understood as the distance from one side to the opposite vertex.

Every triangle has three altitudes (h_a, h_b and h_c), each one associated with one of its three sides. If we know the three sides (a, b, and c) it's easy to find the three altitudes, using the Heron's formula:

Altitudes and Orthocenter

The three altitudes of a triangle (or its extensions) intersect at a point called orthocenter.

The altitude can be inside the triangle, outside it, or even coincide with one of its sides, it depends on the type of triangle it is:

• Obtuse triangle: The altitude related to the longest side is inside the triangle (see h_c, in the triangle above) the other two heights are outside the triangle (h_a, and h_b).
• Right triangle: The altitude with respect to the hypotenuse is interior, and the other two altitudes coincide with the legs of the triangle.
• Acute triangle: all three altitudes lie inside the triangle.

Where is the orthocenter located?

• Obtuse triangle: the orthocenter is outside the triangle.
• Right triangle: the orthocenter coincides with the right angle's vertex.
• Acute triangle: the orthocenter is an inner point.

Altitude of an Equilateral Triangle

The altitude (h) of the equilateral triangle (or the height) can be calculated from Pythagorean theorem. The sides a, a/2 and h form a right triangle. The sides a/2 and h are the legs and a the hypotenuse.

Applying the Pythagorean theorem:

And we obtain that the height (h) of equilateral triangle is:

Another procedure to calculate its height would be from trigonometric ratios.

With respect to the angle of 60º, the ratio between altitude h and the hypotenuse of triangle a is equal to sine of 60º. Therefore:

Altitude of an Isosceles Triangle

The altitude (h) of the isosceles triangle (or height) can be calculated from Pythagorean theorem. The sides a, b/2 and h form a right triangle. The sides b/2 and h are the legs and a the hypotenuse.

By Pythagorean theorem:

And it is obtained that the height h is:

In a isosceles triangle, the height corresponding to the base (b) is also the angle bisector, perpendicular bisector and median.

Altitude of a Right Triangle

In a right triangle the altitude of each leg (a and b) is the corresponding opposite leg. Thus, h_a =b and h_b =a. The altitude of the hypotenuse is h_c.

The three altitudes of a triangle intersect at the orthocenter H which for a right triangle is in the vertex C of the right angle.

To find the height associated with side c (the hypotenuse) we use the geometric mean altitude theorem.

We can calculate the altitude h (or h_c) if we know the three sides of the right triangle.

Download this calculator to get the results of the formulas on this page. Choose the initial data and enter it in the upper left box. For results, press ENTER.

Triangle-total.rar         or   Triangle-total.exe

Note. Courtesy of the author: José María Pareja Marcano. Chemist. Seville, Spain.

Exercise

Find the lengths of the three altitudes, h_a, h_b and h_c, of the triangle ΔABC, if you know the lengths of the three sides: a=3 cm, b=4 cm and c=4.5 cm.

Firstly, we calculate the semiperimeter (s).

We get that semiperimeter is s = 5.75 cm. Then we can find the altitudes:

The lengths of three altitudes will be h_a =3.92 cm, h_b =2.94 cm and h_c =2.61 cm.

Lines Associated with a Triangle

• Media of a Triangle
• Angle Bisector of a Triangle
• Perpendicular Bisector of a Triangle

how to find the altitude of a triangle

Source: https://www.mathematicalway.com/mathematics/geometry/altitude-triangle/

Posted by: dukesligh1984.blogspot.com

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