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Chapter 4: Investigating the Pythagorean Theorem
Frank & Ernest by Bob Thaves
An introduction to the Pythagorean Theorem
The NCTM standards recommends that students investigate and apply the
Pythagorean theorem, as well as create and critique inductive and deductive arguments
concerning the Pythagorean relationship. Specifically, the NCTM Principles and Standards
(2000) recommend the following regarding the study of Pythagorean theorem in the
Geometry and Measurement Standard.
-create and critique inductive and deductive arguments concerning geometric
ideas including the Pythagorean theorem
-draw geometric objects with specified properties, such as side length or angle
measures
-use geometric models to represent and explain numerical and algebraic
relationships
-select and apply techniques and tools to accurately find length, area, volume,
and angle measures to appropriate levels of precision
Let us spark our interest in the Pythagorean theorem with a historical introduction
that shows the power of this theorem. Knowledge of the Pythagorean theorem first dates
back to around 1900 B.C. with the Plimpton 322, a Babylonian mathematical tablet that
contains a table of Pythagorean triples. According to the Chou-pei, an ancient Chinese text,
the Chinese also knew about the Pythagorean theorem many years before Pythagoras and his
followers. Pythagoras provided the first evidence of a written proof of the theorem. He is by
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no means the last to develop proofs for the theorem. Others such as Leonardo da Vinci,
President James Garfield, and Euclid have developed proofs of the Pythagorean theorem.
Pythagoras was born on the Greek island of Samos around 569 B.C. He left Samos about 532
B.C. and traveled to Egypt where he studied until he returned to Croton to found a school for
the study of mathematics, philosophy, and natural science. His school practiced in secret
making it hard to determine Pythagoras’ work from that of his followers. The Pythagoreans,
as they were known, studied numbers, music, and astronomy. They learned in Egypt about a
3 by 4 by 5 rope and how it could be used to make a right triangle. Pythagoras’ interest in
this triangle may have led to his formulation of a proof that bears his name today. He was
able to determine that the sum of the squares of the two shorter sides of a right triangle
always equal the square of the longest side.
Construct a right triangle using a compass and straight edge, label the sides of the
triangle a, b, and c. Construct squares at the sides of your right triangle. Compare the sum of
the area of the squares on the legs with the area of the square on the hypotenuse. State your
conjecture. Discuss your conjecture with other students. What conclusions can you make?
Model your construction with GSP. Discuss your results and if there are any advantages in
using GSP in this way. Notice that you had to construct a square three times, on the two legs
and on the hypotenuse. Is there a quicker way of doing this?
Creating a script tool for a square
Construct a square. In the edit menu, choose “Select All.” In the custom tool, choose
“Create New Tool.” Click on “Show Script View.” What do you notice? Click on “Ok.” You
have made a script tool that you can use from now to construct a square. Discuss the need for
script tools. Construct script tools for your popular figures.
Constructing squares on the sides of any triangle
Open a new sketch, place two points on the sketch. Use your custom square tool to
create a square using the two points (click on each point with the tool). What do you notice?
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Draw a triangle. Use your square tool on the sides of your triangle by selecting two vertices
at a time. What do you notice? Use your custom tools for your popular figures. Give
instances where custom tools will not be useful.
Measuring the areas of the squares
In the sketch above, select the vertices of one of your squares, under the construct
menu, select “Interior.” What do you notice? While the interior is selected, choose “Area”
under the measure menu. Is the measure given reasonable? Why? Repeat this for the
remaining two squares on your triangle. Compare the measures of the areas. Can you make
any conjectures about relationship between the areas?
Label the sides of the triangle a, b, and c. Measure the areas of the squares on your
triangle. In the measure menu, select “Calculate.” Click on area of square on a, then the plus
sign and area of square on b to sum the areas of the squares on the legs a and b. Measure
Angle C. Select the measure of Angle C, the area of the square on the leg c, and the sum of
areas of the squares on the legs a and b. In the Graph menu, select “Tabulate.” What do you
notice? Move the vertices of your triangle and between each move, double click on your
table. What do you notice? To be able to find the relation, you may need to start with angle C
at about 30 degrees and increase to say 120 degrees and adding an entry to the table at
intervals of 10 degrees. Describe your triangle at each interval. What do you notice in your
table when angle C is 90 degrees?
Investigation with equilateral triangles on the sides of the right triangle
Earlier, you made custom tools of your popular figures; is an equilateral triangle one
of them? If not, then construct an equilateral triangle and make a custom tool from your
construction. Use your equilateral triangle tool on the vertices of a triangle ABC so that
equilateral triangles are built on each side of the triangle, exterior to the original triangle. As
in the case for squares above, sum the areas of the equilateral triangles on the sides a and b
and compare with the area of the equilateral triangle on the side c. Make a table of the sum of
areas of the equilateral triangles on the sides a and b and the area of the equilateral triangle
on the side c. As in the case above, vary angle C in intervals of 10 degrees and complete your
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table. What do you notice when angle C is 90 degrees? What is common in the two
situations above, using squares on the sides of a triangle and using equilateral triangles on the
sides? Complete the statement: “When angle C is 90 degrees, …………..” Complete the
statement: “The three squares on the legs of triangle ABC are ….. to each other.” Also, the
three equilateral triangles used on the sides of triangle ABC are …. to each other.” As a hint,
consider relationships such as congruence or similarity.
Make a custom tool of a rectangle with no special conditions such as fixed ratio of
sides. Use your rectangle tool on triangle ABC and follow the investigations above. What do
you notice when triangle ABC is a right triangle? Explain. Complete the statement: “the
rectangles used in this case are ….. to each other.” What conjecture can you make about the
nature of triangle and the relationship between the figures at the sides of the triangle for the
sum of the areas of the figures on the sides a and b to be equal to the area of the figure on the
side c?
Investigation with semi-circles on the sides of the right triangle
Test your conjecture with semi-circles. Repeat above investigations with semi-circles
instead of squares or equilateral triangles. What does this investigation confirm?
Making Generalizations about your findings
State your findings as a theorem. Compare the statement of your theorem with the
statement of the Pythagorean theorem from a high school geometry textbook. Compare and
contrast your theorem with the book theorem. Do you need to modify your theorem? Why?
Do you like your theorem better? Why? How do we know the theorem is true? By now, you
probably have noticed that when your triangle is acute, then c2 < a2 + b2. Justify that when
your triangle is obtuse (angle C > 90º), then the inequality c2 > a2 + b2 holds true. Review the
three generalizations you have formulated. Explain that the first one, c2 = a2 + b2, is called
the Pythagorean theorem.
Dissection Proofs of the Pythagorean Theorem
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The Pythagorean Theorem asserts that for a right triangle with short sides (called
“legs”) of length a and b and long side (called the “hypotenuse”) of length c, then a2+b2=c2.
The theorem is of fundamental importance in Euclidean Geometry where it serves as a basis
for the definition of distance between two points. There are several animated proofs of the
Pythagorean Theorem on the Internet and in your GSP folder. Investigate some of these. In
particular, investigate “Pythagoras.gsp” in your GSP folder. Compare and contrast the 4
cases presented. Summarize four of the proofs. A very simple demonstration starts with a
right triangle and squares on each side, the middle size square is cut through the center and
parallel to the sides of the biggest square. Then the quadrilaterals are hinged and rotated and
shifted to the big square. The smallest square is translated to cover the remaining middle part
of the biggest square. This demonstrates that the sum of the squares on the smaller two sides
equals the square on the biggest side.
The theorem can be proved by first constructing a right triangle ABC.
Figure 4.1: Similar triangle proof of the Pyhtagorean Theorem
In the diagram above, the perpendicular CD splits the right triangle ABC into two pieces,
triangle ACD and triangle CBD, that are similar to the original. Equating the ratios of
hypotenuse to shorter leg in triangle ACD and triangle ABC, we conclude b/x = c/b, it
follows that b2 = cx. Next we use the similarity of triangle CBD and triangle ABC, along
with the fact that DB = AB - AD = c - x, and equate the ratios of hypotenuse to longer leg to
get a/(c - x) = c/a, thus a2 = c(c - x) = c2 – cx, yielding a2 + b2 = (c2 - cx) + cx = c2