Altitude of a Triangle

1 Star2 Stars3 Stars4 Stars5 Stars (No Ratings Yet)
Loading...

Drawing of the three altitudes of a triangle and the orthocenter

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 (ha, hb and hc), 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:

Formula of the three heights of the triangle

Altitudes and Orthocenter

Drawing of the three heights of a triangle and the 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:

Where is the orthocenter located?

Altitude of an Equilateral Triangle

Drawing of the equilateral triangle to calculate its height

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:

Calculation of the height of the equilateral triangle.

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

Formula for the height of the equilateral triangle.

Drawing of the equilateral triangle for the calculation of its height for trigonometric ratios

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:

Calculation of the height of the equilateral triangle for trigonometric ratios.

Altitude of an Isosceles Triangle

Drawing of the isosceles triangle to calculate its altitude

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:

Isosceles triangle height calculation

And it is obtained that the height h is:

Isosceles triangle height formula

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

Drawing the heights of the right triangle

In a right triangle the altitude of each leg (a and b) is the corresponding opposite leg. Thus, ha = b and hb = a. The altitude of the hypotenuse is hc.

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.

Drawing the right triangle for the height theorem

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

Formula of the height by the theorem of the height from the sides

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

Drawing in exercise 1

Find the lengths of the three altitudes, ha, hb and hc, of the triangle Δ ABC, 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).

Calculation of the semiperimeter of a triangle in exercise 1

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

Solution in exercise 1

The lengths of three altitudes will be ha=3.92 cm, hb=2.94 cm and hc=2.61 cm.

Lines Associated with a Triangle


AUTHOR: Bernat Requena Serra

YEAR: 2020


IF YOU LIKED IT, SHARE IT!

Leave a Reply

Your email address will not be published.