Pythagorean Theorem- Lesson 2

The Pythagorean Theorem
Lesson 2

Lesson By: Jonathan Challen

Prerequisites:

Before reaching this lesson, students should:

Know basic triangle theorems (similarity theorems, congruence theorems, corresponding parts theorems, angle sum theorems).
Know how to solve algebraic equations involving squares and square roots.
Have basic knowledge in the use of the computer program, Geometer's Sketchpad.

Objectives:

Discover the Pythagorean Theorem with the aid of a computer acitivity
Explore methods of proving the Pythagorean Theorem
Introduce the Pythagorean Theorem as a method of solving right triangle problems
Find the length of the third side of a right triangle, given the other two sides

Overview:

Previous Day's Review- History of Pythagorean Theorem (5 minutes)
GSP Activity- Discovering the Pythagorean Theorem (20-30 minutes)
Supplemental Activity- Similarity Proof (10 minutes)
Supplemental Activity- Bhaskara's Proof (with animated image) (10 minutes)
Introduce Pythagorean Theorem (explain equation, what variables represent, etc.) (5 minutes)
Example Problems (2 examples) (10 minutes)
Homework/Classwork- Practice Problems Worksheet (10 problems)

Note to Teachers: Supplemental activities are provided if extra time is available. Without the supplemental activities, this entrie lesson can be completed in one 50-minute class period. It is recommended to include the supplemental activities, if time is available.

GSP Activity- Discovering the Pythagorean Theorem:

This activity requires the use of Geometer's Sketchpad, which is a computer program available for both Macintosh and Windows. This activity may also require the use of a calculator to perform simple arithmetic calculations, or students may use the calculator provided with GSP. Teachers should allow students to work independently on this activity, providing only the basic outline of the activity and allowing the students to create their own conjectures regarding the information they find. Teachers should explain that the most important part of the activity is answering the critical thinking questions fully and completely. Teachers should read each step and wait for all students to complete the task before moving on to the next step.

THE GSP SKETCH FOR STUDENTS

THE GSP SKETCH FOR TEACHERS (already has steps completed)

The Activity

Open the file "similar.gsp" in your folder
Measure the following angles: ACB, BDC, ADC
What do these angles have in common? (Answer: They are all 90 degree angles.)
Measure the following angles: BCD, CAD, ACD, CBD
What do you notice about these angles? (Answer: BCD and CAD are equal; CBD and ACD are equal)
Why is this the case? (Answer: Since we know that all of the angles in a triangle add up to be 180 degrees, and that angles ACB, BDC, and ADC are all 90 degrees, angle CAD + angle ACD = 90 and angle BCD + angle CBD = 90. Since angle BCD + angle ACD = 90 (they form angle ACB, which we know is 90 degrees), we can then deduce that angles CBD and ACD are equal and angles BCD and CAD are equal.)
What do these equal angles tell us about triangles CBD and ACD? (Answer: They are similar triangles, by the AA-similarity theorem.)
What does this tell us about these triangles with relation to the large triangle (ABC)? (Answer both of the smaller triangles (CBD and ACD) are similar to the larger triangle (ABC) by AA-similarity.)
Measure the following side lengths: AB, BC, and BD
Do the following computations: BD/BC and BC/AB
What do you notice about these computations? (Answer: They are the same value.)
Why is this the case? (Answer: The side lengths of similar triangles are proportional.)
Measure the following side lengths: AC and AD
Do the following computations: AD/AC and AC/AB
What do you notice about these computations? (Answer: They are the same value.)
Why is this the case? (Answer: The side lengths of similar triangles are proportional.)
Look at these two sets of proportions. Set up two equations based on those proportions. (Answer: BD/BC=BC/AB and AD/AC=AC/AB)
Now solve these equations for BD and AD and do these computations on GSP. (Answer: BD=BC²/AB and AD=AC²/AB)
What do we know about the sum of BD and AD? Do this computation on GSP. (Answer: It equals AB.)
Use this sum to find an equation for AB and do the computation on GSP. (Answer: AB = BC²/AB + AC²/AB)
If we multiply this equation through by AB, what equation do we get? (Answer: AB² = BC² + AC²)
What is the significance of this equation? (It is the Pythagorean Theorem!)
Check the Pythagorean Theorem using your GSP sketch (compute AC² + BC² and AB² and compare).

Similarity Proof:

Similarity Proof

ABC is a right triangle, with the right angle at angle ACB. An altitude is dropped from vertex C, forming right angles at angle CDB and angle CDA. Since all angles in a triangle must sum to 180 degrees, the sum of angle CAD and angle ACD is 90, as is the sum of angle CBD and angle BCD. Therefore, angle CAD must equal angle BCD and angle CBD must equal angle ACD. Thus, we have the angles in triangle ACD and triangle CBD equal. By the AA-similarity theorem, these triangles must be similar. We can then set up the following proportions:

Since p + q = c, we get:

And finally:

Bhaskara's Proof:

Bhaskara Proof

The Bhaskara figure contains a small square within a larger square and forms four right triangles inside the larger square. The length of the sides of the larger square are c, and the lengths of the legs of the right triangles are a and b. An easy deduction leads to the smaller square's sides being b - a. Thus, the area of the larger square can be used to prove the Pythagorean Theorem: