CK-12 Geometry Concepts

http://www.ck12.org Chapter 5. Relationships with Triangles

Orthocenter:The point of concurrency for the altitudes of triangle.

Here is what the orthocenter looks like for the three triangles. It has three different locations, much like the
perpendicular bisectors.

TABLE5.5:

Acute Triangle Right Triangle Obtuse Triangle

The orthocenter is inside the trian-

gle.

The legs of the triangle are two of

the altitudes. The orthocenter is the

vertex of the right angle.

The orthocenter is outside the trian-

gle.

Example A

Which line segment is an altitude of 4 ABC?

In a right triangle, the altitude, or the height, is the leg. If we rotate the triangle so that the right angle is in the lower
left corner, we see that legBCis an altitude.

Example B

A triangle has angles that measure 55◦, 60 ◦,and 65◦. Where will the orthocenter be found?

Because all of the angle measures are less than 90◦, the triangle is an acute triangle. The orthocenter of any acute
triangle is inside the triangle.

Example C

A triangle has an angle that measures 95◦. Where will the orthocenter be found?

Because 95◦> 90 ◦, the triangle is an obtuse triangle. The orthocenter will be outside the triangle.

Watch this video for help with the Examples above.

MEDIA

Click image to the left for more content.

CK-12 Foundation: Chapter5AltitudesB

Vocabulary

Thealtitudeof a triangle, also known as the height, is a line segment from a vertex and perpendicular to the
opposite side.Perpendicularlines are lines that meet at right (90◦) angles. Theorthocenterof a triangle is the
point of concurrency for the altitudes of triangle (the point where all of the altitudes meet).

Guided Practice

1. True or false: The altitudes of an obtuse triangle are inside the triangle.