How to Find the Orthocenter and Altitudes of a Triangle

The altitude and orthocenter of triangle

For a triangle, an altitude is a line that passes through a corner and is perpendicular to the opposite side. Every triangle has three altitudes, one for each corner.

Theory

Orthocenter and Altitudes

The altitudes of a triangle have one common point of intersection. This point is called the orthocenter.

When you want to find the orthocenter of a triangle, you need to construct two of the altitudes. The intersection of these is the orthocenter.

Note! The orthocenter can be outside of the triangle. This happens when one of the angles is greater than $90$ °.

Example 1

A triangle $△ A B C$ has the sides $A B = 7$ , $A C = 5$ and $B C = 6$ . Construct the orthocenter of this triangle.

Before you construct the orthocenter, you need to construct the triangle with the given measurements. Start with the line $A B = 7$ . Set the compass’s radius to 6 and make a faint circle with center $B$ . Then, set the compass’s radius to 5 and make a faint circle with center $A$ . The corner $C$ appears as either of the intersection points between the two circles. Then you end up with the following triangle:

Example of construction of orthocenter of triangle 1

Then, you construct a normal from one corner of the triangle to the opposite side. Repeat this for one of the two remaining corners. What you have done now is to construct two altitudes of the triangle from two of its sides. The orthocenter of the triangle is the intersection between the two altitudes.

Example of construction of orthocenter of triangle 2

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