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.

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.

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.

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.

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.

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

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

- Introduction to constructions
- Copy a line segment
- Sum of n line segments
- Difference of two line segments
- Perpendicular bisector of a line segment
- Perpendicular from a line at a point
- 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)

- 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

- 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 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)

- 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

- 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

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