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

Angle C is obtuse

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.

Try it yourself

Acknowledgements

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

Other constructions pages on this site

• List of printable constructions worksheets

Lines

• Introduction to constructions
• Copy a line segment
• Sum of n line segments
• Difference of two line segments
• Perpendicular bisector of a line segment
• Perpendicular at a point on a line
• Perpendicular from a line through a point
• Perpendicular from endpoint of a ray
• Divide a segment into n equal parts
• Parallel line through a point (angle copy)
• Parallel line through a point (rhombus)
• Parallel line through a point (translation)

Angles

• Bisecting an angle
• Copy an angle
• Construct a 30° angle
• Construct a 45° angle
• Construct a 60° angle
• Construct a 90° angle (right angle)
• Sum of n angles
• Difference of two angles
• Supplementary angle
• Complementary angle
• Constructing  75°  105°  120°  135°  150° angles and more

Triangles

• Copy a triangle
• Isosceles triangle, given base and side
• Isosceles triangle, given base and altitude
• Isosceles triangle, given leg and apex angle
• Equilateral triangle
• 30-60-90 triangle, given the hypotenuse
• Triangle, given 3 sides (sss)
• Triangle, given one side and adjacent angles (asa)
• Triangle, given two angles and non-included side (aas)
• Triangle, given two sides and included angle (sas)
• Triangle medians
• Triangle midsegment
• Triangle altitude
• Triangle altitude (outside case)

Right triangles

• Right Triangle, given one leg and hypotenuse (HL)
• Right Triangle, given both legs (LL)
• Right Triangle, given hypotenuse and one angle (HA)
• Right Triangle, given one leg and one angle (LA)

Triangle Centers

• Triangle incenter
• Triangle circumcenter
• Triangle orthocenter
• Triangle centroid

Circles, Arcs and Ellipses

• Finding the center of a circle
• Circle given 3 points
• Tangent at a point on the circle
• Tangents through an external point
• Tangents to two circles (external)
• Tangents to two circles (internal)
• Incircle of a triangle
• Focus points of a given ellipse
• Circumcircle of a triangle

Polygons

• Square given one side
• Square inscribed in a circle
• Hexagon given one side
• Hexagon inscribed in a given circle
• Pentagon inscribed in a given circle

Non-Euclidean constructions

• Construct an ellipse with string and pins
• Find the center of a circle with any right-angled object