Why do this problem?
This problem shows an approach to proving Pythagoras'
Theorem by equating areas calculated in different ways, giving
the opportunity to practise calculating areas. There is
historical interest to this proof as it was discovered by James
A Garfield who went on to become President of the United
States.
Possible approach
Show the image of the trapezium made up of two identical
right-angled triangles joined as shown in the problem. Ask
students to visualise the shape formed by rotating a copy of
the trapezium about the centre point marked. They could
sketch the resulting shape and share their answers in pairs.
Then rotate the trapezium to verify their answers.
Discuss ways of working out the area of the trapezium, both
by calculating areas of squares and right-angled triangles.
Set each pair to work the area out in two different ways, one
by finding the area of the square and halving it, and the
other by summing the areas of the three right-angled
triangles.
Each pair can be given the challenge, once they have two
expressions for the same area, of producing a poster or
presentation to explain how this leads to Pythagoras'
Theorem. The lesson could end with each pair presenting their
proof to the rest of the class.
Key questions
How can we be sure that the green triangle is a right-angled
triangle?
What shape is made by rotating the trapezium around the
centre of the longest side?
How could we work out the area of the trapezium? Is there
more than one way?
Possible extension
Explore other proofs of Pythagoras in
this problem.
Possible support
Tilted Squares offers another way to think about proving
Pythagoras's Theorem.