Altitude of a triangle (outside case)

This page shows how to construct one of the three possible altitudes of a triangle, using only a compass and straightedge or ruler. The other two can be constructed in the same way.

An altitude of a triangle is a line which passes through a vertex of a triangle, and meets the opposite side at right angles. For more on this see Altitude of a Triangle.

The three altitudes of a triangle all intersect at the orthocenter of the triangle. See Constructing the orthocenter of a triangle.

Method

The construction starts by extending the chosen side of the triangle in both directions. This is done because, this being an obtuse triangle, the altitude will be outside the triangle, where it intersects the extended side PQ. After that, we draw the perpendicular from the opposite vertex to the line. This is identical to the construction A perpendicular to a line through an external point. Here the 'line' is one side of the triangle, and the 'external point' is the opposite vertex.

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
This case is shown on the companion page Altitude of a triangle.

However, if one of the angles opposite the chosen vertex is obtuse, then it will lie outside the triangle, as below.

 Angle C is obtuse

The angle ACB is opposite the chosen vertex A, and is obtuse (greater than 90°). and is the reason the first step of the construction is to extend the base line, just in case this happens. The altitude meets the extended base BC of the triangle at right angles.

Printable step-by-step instructions

The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

Proof

The proof of this construction is trivial. This is the same drawing as the last step in the above animation.

Argument Reason
1 The segment SR is perpendicular to PQ Created using the procedure in Perpendicular to a line through an external point. See that page for proof.
2 The segment SR is an altitude of the triangle PQR. From (1) and the definition of an altitude of a triangle (a segment from the a vertex to the opposite side and perpendicular to that opposite side).

- Q.E.D

Try it yourself

Click here for a printable construction worksheet containing two 'altitude of a triangle' problems to try. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.

Acknowledgements

Thanks to Aaron Strand of Carmel High School, Indiana for suggesting, reviewing, and proofreading this construction