The Pythagorean Theorem
Definitions: The side opposite the right angle in a right triangle
is called the hypoteneuse. (This term applies only to right triangles.)
The two other sides (those forming the right angle) are called legs.
Often the legs are labeled a and b and the hypoteneuse is labeled c. Usually
the angles opposite the sides of a triangle have the same letters, but
capitalized: A, B, and C.
Group problems
A. First activity
With drinking straws and paper clips (as described in a previous lesson),
or ruler and compass, or this interactive diagram,
make a triangle with one side of length 9 cm, one side of length 12 cm,
and one side of adjustable length.
Make at least 6 triangles, with the length of the adjustable side ranging
from the shortest possible to the longest possible. (Use the shortest and
longest theoretical sides, not just the ones you can make with your straws.)
Call the sides a (=9), b (=12), and c; call the angles opposite them A,
B, and C, respectively. Make a table
that shows the lengths of the sides, the measures of the angles, and two
calculated values for your triangles you constructed; use these headings
for the columns of your table:
a, A, b, B, c, C, a2+b2, c2 Arrange
the rows so that c is increasing.
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What is the shortest theoretical length, and the corresponding angle? Don't
restrict yourself to whole numbers.
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What is the longest theoretical length, and the corresponding angle?
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Is it possible to get a right angle for C? If so, make sure one is on your
table.
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Make a conjecture relating the length of the adjustable side c to the measure
of the opposite angle C.
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Make a conjecture relating a2+b2, c2,
and the measure of angle C.
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What is the sum of the angles A and B in any right triangle (where
C is the right angle)? Why?
B. Second activity 
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Construct (any way you want) a right triangle on paper. Call its legs a
and b, and its hypoteneuse c.
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Make a total of 4 copies of this triangle and cut them out.
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Arrange the triangles so that the longer leg of the second triangle continues
the shorter side of the first triangle, and so on. Angle A on the first
triangle should meet angle B on the second triangle. You will get a quadrilateral
with a quadrilateral hole in the middle. (If it doesn't look square, compare
with the picture at right.)
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Prove that the outer quadrilateral is a square. Hints:
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What is the length of each side of the quadrilateral?
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What is the measure of each angle of the quadrilateral?
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Prove that the inner quadrilateral is a square. Hints:
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What is the length of each side of the quadrilateral?
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What is the measure of each angle of the quadrilateral?
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Compute the area of the outer quadrilateral (square) using the length of
its side. Do this both for actual measurements and symbolically using a,
b, and c.
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Compute the area of the outer quadrilateral by adding the areas of the
four triangles and the area of the inner square. Do this both for actual
measurements and symbolically using a, b, and c.
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Are the two numerical areas equal? If not, you did something wrong.
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Write a symbolic equation between the two expressions for the area of the
outer square. Use algebra to manipulate the equation to conclude that a2+b2
= c2
The Pythagorean Theorem.
(a) In a right triangle with legs a and b and hypoteneuse c,
a2+b2 = c2
(b) If a triangle has sides a, b, and c, and a2+b2
= c2 then the triangle is a right triangle with hypoteneuse
c.
C. Check out these proofs of the
Pythagorean Theorem on the Web in the computer lab. People with more computer
experience help those with less.
Individual homework:
A. Choose one of the proofs of the Pythagorean Theorem on the Web and
make a paper-and-scissors version. That is, make a worksheet with instructions
telling what to cut out, and a clue of what the result should look like.
Include a separate answer sheet showing what the final result should be.
B. Cool Stuff to Make with Right Triangles.
To the Math 129 home page
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