## Reflection: Coherence Altitude to the Hypotenuse - Section 3: Generalizations

In years past, I had skipped the lesson on the geometric mean because it was hardly ever tested on state tests. Having now taught the CCSS version of the Geometry course twice, I can't see how I ever skipped it.

The geometric mean, particularly as related to the situation when the altitude is drawn to the hypotenuse of a right triangle, is a concept that surfaces at many points in the geometry course. In other words, it has endurance.

For example, we use it in the proof of the Pythagorean Theorem using similar triangles. We use it again to prove the slope criteria for perpendicular lines. We use it also in construction problems. For example, the one that asks students to construct a square with the same area as a given rectangle.

I find that it is important to find these types of topics/concepts that have endurance because these are what send the message to students that they will use what they are learning. It will be useful to them in the future. It is not to be learned for a test and then discarded.

Looking for concepts with endurance
Coherence: Looking for concepts with endurance

# Altitude to the Hypotenuse

Unit 6: Similarity
Lesson 6 of 8

## Big Idea: There's more to right triangles than the Pythagorean Theorem.

Print Lesson
Standards:
Subject(s):
Math, Proofs, right triangle, altitudes
80 minutes

### Anthony Carruthers

##### Similar Lessons

###### Applying Triangle Congruence
Geometry » Triangles and Congruence
Big Idea: Students learn how proving triangles congruent can be used as part of a larger strategy in a proof.
Favorites(4)
Resources(18)
Ault, CO
Environment: Rural

###### Time for Triangles
Geometry » Tremendous Triangles
Big Idea: Students will measure, cut and measure again to learn and prove key properties of triangles
Favorites(4)
Resources(28)
Saratoga Springs, NY
Environment: Suburban

###### Prove It (Part 1)
Geometry » Similar and Right Triangles
Big Idea: Students work on proving hypotheses – and end up proving the Pythagorean Theorem.
Favorites(1)
Resources(11)
Amsterdam, NY
Environment: Urban