© 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.