
Altitude of a triangle
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 the side may not be long enough for later steps to work.
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 
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.
This case is demonstrated on the companion page
Altitude of an triangle (outside case),
and is the reason the first step of the construction is to extend the base line, just in case this happens.
Printable stepbystep instructions
The above animation is available as a
printable stepbystep 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
Other constructions pages on this site
Lines
Angles
Triangles
Right triangles
Triangle Centers
Circles, Arcs and Ellipses
Polygons
NonEuclidean constructions
(C) 2011 Copyright Math Open Reference. All rights reserved

