Lesson 7A Proof of the Pythagorean Theorem
Learning Goal
Let’s prove the Pythagorean Theorem.
Learning Targets
I can explain why the Pythagorean Theorem is true.
Lesson Terms
- hypotenuse
- legs
- Pythagorean Theorem
Warm Up: Notice and Wonder: A Square and Four Triangles
Problem 1
What do you notice? What do you wonder?
Activity 1: Adding Up Areas
Problem 1
Both figures shown here are squares with a side length of
What is the total area of each figure?
Find the area of each of the 9 smaller regions shown the figures and label them.
Add up the area of the four regions in Figure F and set this expression equal to the sum of the areas of the five regions in Figure G. If you rewrite this equation using as few terms as possible, what do you have?
Are you ready for more?
Problem 1
Take a 3-4-5 right triangle, add on the squares of the side lengths, and form a hexagon by connecting vertices of the squares as in the image. What is the area of this hexagon?
Activity 2: Let’s Take It for a Spin
Problem 1
Find the unknown side lengths in these right triangles.
Activity 3: A Transformational Proof
Problem 1
Use the applets to explore the relationship between areas.
Consider Squares
and . Check the box to see the area divided into five pieces with a pair of segments.
Check the box to see the pieces.
Arrange the five pieces to fit inside Square
. Check the box to see the right triangle.
Arrange the figures so the squares are adjacent to the sides of the triangle.
If the right triangle has legs
and and hypotenuse , what have you just demonstrated to be true? Try it again with different squares. Estimate the areas of the new Squares,
, , and and explain what you observe. Estimate the areas of these new Squares,
, , and , and then explain what you observe as you complete the activity. What do you think we may be able to conclude?
Print Version
Your teacher will give your group a sheet with 4 figures and a set of 5 cut out shapes labeled D, E, F, G, and H.
Arrange the 5 cut out shapes to fit inside Figure 1. Check to see that the pieces also fit in the two smaller squares in Figure 4.
Explain how you can transform the pieces arranged in Figure 1 to make an exact copy of Figure 2.
Explain how you can transform the pieces arranged in Figure 2 to make an exact copy of Figure 3.
Check to see that Figure 3 is congruent to the large square in Figure 4.
If the right triangle in Figure 4 has legs
and and hypotenuse , what have you just demonstrated to be true?
Lesson Summary
The figures shown here can be used to see why the Pythagorean Theorem is true. Both large squares have the same area, but they are broken up in different ways. (Can you see where the triangles in Square G are located in Square F? What does that mean about the smaller squares in F and H?) When the sum of the four areas in Square F are set equal to the sum of the 5 areas in Square G, the result is
This is true for any right triangle. If the legs are
The grid can be used to create a right triangle, where the line segment is the hypotenuse and the legs measure 24 units and 7 units:
Since this is a right triangle,