© 2011 by Mathematics Assessment Page 1 Proofs of the Pythagorean Theorem?
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Proofs Of The Pythagorean Theorem?
Here are three attempts to prove the Pythagorean theorem.
Look carefully at each attempt. Which is the best 'proof' ?
Explain your reasoning as fully as possible.
Attempt 1:
Suppose a right triangle has sides of length
a, b and c
Draw squares on the three sides as shown.
Divide these squares into smaller squares.
You can see that the number of squares on the two shorter
sides add up to make the number of squares on the longest
side.
So: a2+b2=c2

a
b
c
a
b

c

Attempt 2
Suppose that you start with four
right triangles with sides of length
a, b and c and a square tray with
sides of length a+b. a
b
c

a+b

a+b TRAY

You can arrange the triangles into
the tray in two different ways as
shown here.
In the first way, you leave two
square holes. These have a
combined area of a2 + b2.
In the second way you leave one
large square hole. This has an area
of c2.
Since these areas are equal
a2 + b2 = c2

a b
c b c

a
b a
c

c

a

b

c
a b

c

Copyright © 2011 by Mathematics Assessment Page 2 Proofs of the Pythagorean Theorem?
Resource Service. All rights reserved.
Attempt 3:
The proof of the Pythagorean theorem is clear from this
diagram.
The squares on the two shorter sides of the black triangle
are each made from two congruent triangles.

These fit together to make the square on the longest side-
the hypotenuse.