Chapter 11 Project
In this chapter we have applied the Pythagorean Theorem to many different applications that we may encounter in our daily lives. In this project, we will explore other extensions of this very important theorem.
Part I
This activity is designed to illustrate why the Pythagorean Theorem is true. It is based on the actual proof that is attributed to Pythagoras.
The following figure illustrates the fact that 
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Figure 11.6
1. Determine the area of each of the squares.
2. According to the Pythagorean theorem, the sum of the areas of the smaller two squares is
equal to the area of the larger square. Write an equation for this statement using the areas
you found in exercise 1. Your equation should be the Pythagorean theorem.
3. Let a = 6, b = 8, and c = 10. Calculate the area of the squares, and verify that the Pythagorean
theorem is correct for these values.
4. Draw Figure 11.6 on graph paper, using the values a = 6, b = 8, and c = 10. Cut up and reassemble
the two small squares to form the larger square.
Part II
An iterative example of the using the Pythagorean Theorem is demonstrated in the following figure:
Determine the exact length for x1, then x2, then x3, and so on until you get a value for x6. Do not approximate the lengths.
Part III
The equation x2 + y2 = z2, where x, y, and z are integers, is called a Diophantine equation. A Greek mathematician, Diophantus of Alexandria, proved that any set of three integers that satisfy this
equation has the form
x = a2 - b2, y = 2ab and z = a2 + b2,
where a and b are integers.
This Diophantine equation is a special case of the Pythagorean theorem that was used in the chapter to solve application problems. We will examine such equation further.
1. Complete the following table of values for the previous Diophantine equation:
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Case |
a |
b |
x = a2 - b2 |
y = 2ab |
z = a2 + b2 |
Diophantine triple |
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1. |
2 |
1 |
3 |
4 |
5 |
3, 4, 5 |
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2. |
3 |
1 |
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3. |
3 |
2 |
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4. |
4 |
1 |
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5. |
4 |
2 |
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6. |
4 |
3 |
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7. |
5 |
1 |
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8. |
5 |
2 |
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9. |
5 |
3 |
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10. |
5 |
4 |
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2. Check each row of the table to be sure that the Diophantine triple does in fact satisfy the Pythagorean theorem. Summarize what your check revealed.
3. Which of the cases have Diophantine triples that are just multiples of a smaller Diophantine triple. What does this observation indicate to you about finding Diophantine triples that satisfy the Pythagorean theorem?
4. How many different Diophantine triples do you think can be found which satisfy the Pythagorean theorem?
5. Substitute the Diophantine expressions for x, y, and z into the Pythagorean theorem to see if they satisfy the theorem.
6. Program your calculator to create Diophantine triples:
PROGRAM:DIOPHANT
Disp "A?,A>1"
Input A
seq(A,X,1,A-1,1)
L1
seq(X,X,1,A-1,1) 
L2
L12-L22
L3
2*L1*L2
L4
L12+L22
L5
Disp "PRESS STAT 1"
Disp "TO VIEW L3,L4,L5"
7. Use the program from step 6 to create Diophantine triples when a = 10 and b assumes integer values from 1 to 9.
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a |
b |
x, L3 |
y, L4 |
z, L5 |
Diophantine triple |
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10 |
1 |
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10 |
2 |
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10 |
3 |
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10 |
4 |
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10 |
5 |
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10 |
6 |
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10 |
7 |
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10 |
8 |
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10 |
9 |
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8. Check whether all the triples in step 7 satisfy the Pythagorean theorem. Report the results of your check.
Part IV
To finish our examination of the Pythagorean theorem and its related Diophantine triples, research the mathematicians, Pythagoras of Samos or Diophantus of Alexander in the library or using the Internet. Write a one-page summary of interesting facts that you discover about either of these famous mathematicians.