3 Altitudes Of A Triangle

Altitude of a triangle

This page shows how to construct one of the three possible altitudes of a triangle, using simply a compass and straightedge or ruler. The other two tin exist 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 than on this see Altitude of a Triangle.

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

Method

The structure starts past extending the chosen side of the triangle in both directions. This is done because the side may not exist 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 signal. Here the 'line' is one side of the triangle, and the 'external point' is the opposite vertex.

It can exist exterior the triangle

In most cases the altitude of the triangle is within 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 beneath. The angle ACB is contrary the called vertex A, and is obtuse (greater than ninety°).

Bending C is birdbrained

The altitude meets the extended base BC of the triangle at correct angles. This case is demonstrated on the companion page Distance of an triangle (outside example), and is the reason the first pace of the construction is to extend the base of operations line, just in instance this happens.

Printable step-by-step instructions

The in a higher place blitheness is available as a printable pace-by-stride instruction canvas, which can be used for making handouts or when a estimator is not available.

Proof

The proof of this structure is trivial. This is the same cartoon as the concluding step in the higher up animation.

	Argument	Reason
1	The segment SR is perpendicular to PQ	Created using the procedure in Perpendicular to a line through an external point. Meet 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 contrary side).

- Q.Due east.D

Attempt it yourself

Click here for a printable construction worksheet containing 2 'altitude of a triangle' problems to endeavor. 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 structure

Other constructions pages on this site

List of printable constructions worksheets

Lines

Introduction to constructions
Copy a line segment
Sum of due north line segments
Departure of 2 line segments
Perpendicular bisector of a line segment
Perpendicular at a betoken on a line
Perpendicular from a line through a point
Perpendicular from endpoint of a ray
Divide a segment into north equal parts
Parallel line through a point (bending re-create)
Parallel line through a point (rhombus)
Parallel line through a point (translation)

Angles

Bisecting an angle
Re-create an bending
Construct a thirty° bending
Construct a 45° angle
Construct a 60° angle
Construct a 90° bending (correct bending)
Sum of n angles
Difference of two angles
Supplementary bending
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 noon angle
Equilateral triangle
30-60-90 triangle, given the hypotenuse
Triangle, given 3 sides (sss)
Triangle, given ane side and adjacent angles (asa)
Triangle, given 2 angles and non-included side (aas)
Triangle, given ii sides and included angle (sas)
Triangle medians
Triangle midsegment
Triangle altitude
Triangle altitude (outside case)

Right triangles

Correct Triangle, given one leg and hypotenuse (HL)
Right Triangle, given both legs (LL)
Correct Triangle, given hypotenuse and 1 angle (HA)
Correct Triangle, given one leg and 1 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 signal
Tangents to two circles (external)
Tangents to ii 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 circumvolve

Not-Euclidean constructions

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