An Investigation of
Altitudes in a Triangle

Background

In this section of the exploration you will be asked to discover the way the altitudes of a triangle relate to each other. As you work through the material, you may be asked to write down the answers to the questions in the investigation or to save the information and your sketch on a disk. Ask your teacher for specific directions.

Exploring the Altitudes of a Triangle

Click on the image below to activate the Geometer's Sketchpad Triangle Sketch. You are going to use this sketch to construct the altitudes of a triangle. The instructions for the construction are below the picture. Following the construction, you will use your sketch to investigate relationships involving the altitudes.

Construct

Construct the altitude to side AC through vertex B.
Directions for construction of an altitude.
Script for constructing an altitude.
Construct a second altitude in the triangle through vertex A.
Make a point at the intersection of these two altitudes.
Construct the third altitude through vertex C.

Investigate

When you constructed the third altitude, did it pass through the same point as the other two altitudes?
Use the selection tool (the arrow) to move the vertices of the original triangle. Does the way the altitudes intersect change?
Write a conjecture about the way the altitudes intersect in a triangle. (Check your answer.)
Move one vertex of the triangle so your triangle is an acute triangle. Where is the intersection of the altitudes?
Now move the vertex to make your triangle a right triangle. What happens to the intersection of the altitudes in this case?
Now make your triangle an obtuse triangle. What happens to the intersection of the altitudes now? (Check your answers to 4, 5 and 6.)

We call the intersection of the altitudes of a triangle the orthocenter of the triangle.

Before you leave this investigation....

Did you follow your teacher's directions about saving your work and turning it in?