The Pythagorean Theorem
Suppose we have a right triangle with vertices labeled A, B, and C as in the diagram below. Note that the
angles are also labeled A, B, and C. The square symbol at angle/vertex C indicates that it is a right angle.
B
c
a
A
b
C
By convention the lengths of the sides opposite the vertices are labeled a, b, and c.
The full theorem makes two statements. First, if a triangle has a right angle, then the sum of the squares of
the lengths of the two shorter sides equals the square of the length of the longer side. In symbols,
2
2
2
if C is a right angle, then a + b
= c .
Second, if sum of the squares of the lengths of any two sides equals the square of the length of the remaining side of
a triangle, then the triangle is a right triangle. Again symbolically,
2
2
2
if a + b
= c , then angle C is a right angle.
Taken together, the Pythagorean Theorem establishes an equivalence between the measure of an
angle and a relationship amongst the lengths of the sides of a triangle. At first glance this might seem no
more reasonable than asserting that there is a profound and unvarying connection between the color of a
car and the material the seats are made of.
Questions.
1.)
What is a right angle?
Give other names for a right angle. Speculate on why it is called a right angle. The German adjective to
describe such an angle is Recht, which carries with it approximately all of the other very profound
meanings of “right. Weird. If you can name this angle in any other language and give any other meanings
of the particular name, please do!
2.)
What makes right angles so special? How are they used?
Right angles have been studied by all civilizations in all times. Speculate on the reasons for this. What
universal experience/fact of nature might lead to the right angle idea?
3.)
Is the theorem believable?
The Egyptians and Babylonians used the theorem without having a formal proof of it. Or at least we have
no record of such proofs. Even without proof, is the theorem supported by data? We will examine and
measure the sides of triangles that we reasonably believe have a right angle and study the relationship
between these lengths. Record your data on the sheet on the opposite side.
Right Triangle Data.
We will numerically investigate the Pythagorean Theorem by finding triangles that we believe to be right
triangles, measuring their sides, and testing the theorem’s conclusion on the data.
In addition to the measurements supply the following information for each triangle:
a physical description of the triangle, e.g., half of a sheet of notebook paper;
your measuring tool;
any special methods used to make the measurement – we should be able to reproduce your procedure very
closely;
your measurements must be metric - you may measure the same length more than once to be sure you have
a reliable measurement;
one of the triangles you measure must be large enough to require the tape.
leg
a
leg
b
hypotenuse
2
c
a
differnce
+ b
2
c
2
a
2
+ b
2
–c
% difference
2
100 * (a
2
)/c
+ b
2
–c
2
2
i.
ii.
iii.
iv.
v.
i.
ii.
iii.
iv.
v.
4.)
Note that some of your numbers in the "difference" and "%difference" columns may be negative. What
does the negative sign mean in this case?
5.)
Give an hypothetical example of a situation that produces a small difference but a large %difference.
6.)
Give an hypothetical example of a situation that produces a large difference but a snall %difference.