Explore the relationship between the length of the altitude to the hypotenuse in a right triangle and the lengths of the two segments formed when this altitude intersects the hypotenuse.

• Students will investigate and state the relationship between the altitude to the hypotenuse and the two segments of the hypotenuse formed by this altitude.
• Students will write a conditional statement representing this relationship.

• altitude (height of triangle)
• hypotenuse
• geometric mean

This lesson involves observing changes in a construction of a right triangle. Students will progress from manipulating objects, describing observations, and inferring relationships, to making deductive arguments to state a theorem. As a result, students will:

• Discover the relationship between the altitude to the hypotenuse and the two segments on the hypotenuse cut by this altitude.
• Discover that the length of the altitude is the geometric mean between the lengths of the two segments forming the hypotenuse.