The
Pythagorean Theorem states that in a right triangle with legs of lengths *a*
and *b* and hypotenuse of length *c,* _{} (See Figure1a.)
The theorem appears as Proposition 47 of Book I of Euclid's *Elements.* Euclid attributes it to
Pythagoras(approximately 580-500 BCE), but it almost certainly was known much
earlier to Egyptian, Mesopotamian, Chinese, and Indian mathematicians.

Figure 1a Figure 1b

You
undoubtedly have used the Pythagorean Theorem as a tool for finding the length
of the third side of a right triangle when the lengths of two sides are given.
Notice, however, that the Pythagorean Theorem is a statement about squares,
namely *a*^{2}, *b*^{2}, and *c*^{2}. It
shouldn't surprise you, then, that Euclid and his predecessors thought of the
Pythagorean Theorem as a statement about areas of squares and visualized it as
in Figure 1b.

Figure 2

The
diagram in Figure 2 of the special case of the 3-4-5
right triangle appeared in the Chinese text, *Nine Chapters on the
Mathematical Art,* which was originally composed in about200 BCE and
rewritten and expanded in about 250 CE by Liu Hui. Like Euclid's *Elements,* the *Nine Chapters* was a
compilation of the mathematics known to that day (Swetz and Kao, 17, 26-27). In
fact, the Pythagorean Theorem appears in an older Chinese text, *The
Arithmetical Classic of the Gnomon and the Circular Paths of Heaven,*
probably written during the sixth century BCE, the time of Pythagoras, and
consisting of results known up to that time (Swetz and Kao, 14).

Many
ancient proofs of the Pythagorean Theorem employ geometric algebra; that is,
they rely on finding the area of a given figure in two different ways. Figure
3illustrates one such proof. Viewing the figure as a square with side length *c,*
the area of the figure is *c*^{2}. On the other hand, dissecting the
figure into four right triangles and one small square, as shown, yields area

_{}

Equating these two formulas for the area of the large square, we obtain

_{}

or _{}

or _{}

This proof often
is credited to the Indian mathematician Bhaskara (1114-1185 CE), but it
appeared much earlier in the Chinese text, *The Arithmetical Classic of the
Gnomon and the Circular Paths of Heaven* (Swetz and Kao, 13-15).

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**1.** Use
the Pythagorean Theorem to find the length of the third side of the right
triangle.

**1a.**
**1b.**

**1c.**

**2.** The
converse of the Pythagorean Theorem says: * If a triangle has sides of
lengths a, b, and c and a*^{2}* + b*^{2}* = c*^{2},*
then the triangle is a right triangle with legs of lengths a and b and
hypotenuse of length c.* Use the Pythagorean Theorem and its converse to
determine whether or not each triangle is a right triangle.

**2a.**
**2b.**

**2c.**

**3**. Here
is another proof of the Pythagorean Theorem that uses geometric algebra. Complete
the proof by writing the area of the large square shown in Figure 4 in two
different ways. Then equate your two formulas for the area of the square and
simplify to obtain _{} A version of this proof
appears in the Chinese text,* Arithmetical Classic of the Gnomon and the
Circular Paths of Heaven.*

Figure4

**4.** Here
is a proof of the Pythagorean Theorem that uses a trapezoid rather than a
square. Complete the proof by writing the area of the trapezoid shown in
Figure5 in two different ways. Then equate your two formulas for the area of the
trapezoid and simplify to obtain _{} Recall that the area *A*
of a trapezoid with parallel sides of lengths *a* and *b* and height *h*
is given by *A =*(1/2)(*a* + *b*)*h.*

Figure 5

This
proof of the Pythagorean Theorem is due to President James A.
Garfield(1831-1881), who published it in the *New England Journal of
Education* in 1876 (Dunham Problems, 8), before he was elected President of
the United States in 1880. Garfield took office in January of 1881 and was
assassinated later that year.

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**Instructor Notes**

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**Objective:** Students will verify the Pythagorean
Theorem by computing areas of geometric figures, as mathematicians throughout history
have done.

**How to Use:** Have students work on this activity
individually or in small groups. Before students work on Problems 3 and 4, go
over the example accompanying Figure 3 with them, making sure they understand
the technique of writing the area of a figure in two different ways.

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**Solutions:
1a.** 13 **1b.**
15 **1c.**
30

**2a.** Yes,
by the converse of the Pythagorean Theorem

**2b.** No,
by the Pythagorean Theorem

**2c.** No,
by the Pythagorean Theorem

**3. ** See Figure
4. Viewing the figure as a square with side length _{} the area of the
figure is _{}. On the other hand,
dissecting the figure into four right triangles and one square, as shown,
yields area _{} Equating these two
formulas for the area of the large square, we obtain

_{}

or _{}

or _{}

**4.** See
Figure5. Viewing the figure as a trapezoid with parallel sides of lengths *a*
and *b* and height _{} the area of the figure
is _{} On
the other hand, dissecting the figure into three right triangles, as shown,
yields area _{} Equating these two
formulas for the area of the trapezoid, we obtain

_{}

or _{}

or _{}

**References:** Activity from *Lengths, Areas, and
Volumes, *by J. Beery, C. Dolezal, A. Sauk, and L. Shuey, in *Historical
Modules for the Teaching and Learning of Secondary Mathematics,*
Mathematical Association of America, Washington, D.C., 2003.

Dunham, William,
*Problems for Great Theorems* (unpublished problem sets to accompany *Journey
Through Genius: The Great Theorems of Mathematics*),Muhlenberg College, Allentown, Pennsylvania, 1994.

Swetz, Frank J.,
and T.I. Kao, *Was Pythagoras Chinese?: An Examination of Right Triangle
Theory in Ancient **China**,* National Council of Teachers of Mathematics, Reston, Virginia, 1977.

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