How to construct (draw) one of the three altitudes of a triangle (original) (raw)
Altitude of a triangle
This page shows how to construct one of the three possible altitudesof 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. SeeConstructing 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 constructionA 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 pageAltitude 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 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
Other constructions pages on this site
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
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
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