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'Pythagoras Proofs' printed from http://nrich.maths.org/
Here are three different ideas which can lead to proofs of
Pythagoras' Theorem. Can you make sense of them? Which proof do you
find most "convincing"? Which do you find easiest to understand?
Which would you find easiest to explain to someone else?
Draw a square and mark a point a fixed distance from each
corner. Join from each of these points to the nearest corner on the
opposite side as in the left hand drawing. Cut out the pieces
marked in blue and rearrange them to make an L shape like the one
on the right.
Use your cutouts to prove Pythagoras' Theorem.
This idea is explored in more detail
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Rotate a copy of the trapezium about the centre of the longest side
of the blue triangle to make a square. What is the area of the
square? From this formula for the area of this square derive a
formula for the area of the trapezium. Now write down the area of
the trapezium as the sum of the areas of the three right angled
triangles. Use these results to give a proof of Pythagoras' Theorem
explaining each step.
You can find out more about this proof
Take any right-angled triangle and label its sides $a,b$ and
Enlarge it by scale factor $a$ to make the red triangle and by
scale factor $b$ to make the blue triangle.
Join the two enlargements together as shown.
Show that the two joined enlargements form an enlargement of the
original triangle by scale factor $c$, and use this to prove
You can explore this idea more