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Session 6, Part B: Proving the Pythagorean Theorem
 
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Session 6, Part B:
Proving the Pythagorean Theorem

In This Part: What Is a Theorem? | Constructing a Proof | More Proofs

For the proof outlined below, follow the directions at each step, and answer the questions as you work. When you are finished, you will have constructed a proof of the Pythagorean theorem. Note 2

Step 1: Construct an arbitrary right triangle that is not isosceles. Label the short leg a, the long leg b, and the hypotenuse c.

The proof still works if the triangle is isosceles -- that is if a = b -- but it is always better to work through a proof with an example that is not special in any way.

Step 2: Construct two squares whose sides have length a + b.

Step 3: Dissect one of the squares as shown below:

This dissection yields a square with side length a in one corner, a square with side length b in the opposite corner, and two rectangles, each cut along one diagonal into two right triangles.

Problem B1

Solution  

Show that each of the four triangles you have just created is congruent to the original right triangle.

 
 

Step 4: Dissect the other square as shown below:

This dissection yields four triangles congruent to the original right triangle and a remaining piece in the center.

 

Problem B2

Solution  

Show that the piece in the center is a square with side length c.

Stop! Do the above problem before you proceed. Use the tip text to help you solve the problem if you get stuck.
Try to show that the piece's angles must all be right angles.   Close Tip

 
 

Step 5: The two original squares have the same area.

The eight triangles are congruent. So the four from the first square are equal in area to the four from the second square.

Shaded Area Here

Equals

Shaded Area Here

Step 6: Remove the four triangles from each square. What remains in the first square will have the same area as what remains in the second square.

Unshaded Area Here

Equals

Unshaded Area Here

The geometric equality -- the Pythagorean theorem, as Euclid knew it -- has been shown.

Our more modern algebraic interpretation, a2 + b2 = c2, follows from the algebraic formulas for the areas of the squares. The areas of the two squares on the left are a2 and b2. The area of the square on the right is c2. Geometric reasoning tells you that the areas on the left (a2 + b2) and right (c2) are equal.


video thumbnail
 

Video Segment
In this video segment, the participants help demonstrate why a2, b2, and c2 are all squares with 90° angles at their vertices.

Was your method similar to the one shown here? If not, in what ways was it different?

If you are using a VCR, you can find this segment on the session video approximately 10 minutes and 51 seconds after the Annenberg Media logo.

  

 

Part B: Constructing a Proof, Steps 1-6 adapted from Connected Geometry, developed by Educational Development Center, Inc. pp. 198, 199, 200. © 2000 Glencoe/McGraw-Hill. Used with permission. www.glencoe.com/sec/math

Next > Part B (Continued): More Proofs

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