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Two Pythagorean puzzles
Number 1

This is all based on a right-angled triangle and the squares sitting on each of the three sides.

The centre point of the second largest square is found, and lines are drawn through that point which are parallel to the sides of the largest square.

Then the aim is to fit the smallest square and the pieces of the middle-sized square into the largest square.

By printing out the diagram below, you can see whether you can make them fit.

To see this puzzle demonstrated you can click here.

What does this tell you about:

  • the relationship between the areas of the three squares?
  • the relationship between the lengths of the three sides of the triangle?

This is adapted from the Adult Numeracy Teaching (ANT) course and materials from the ANTOnline site (currently under development)

You can visit here to get instruction for making your own Pythagorean puzzle.

Number 2 - The kou ku theorem
Follow the directions below to produce another proof of the Pythagorean (or kou ku) Theorem based on a Chinese method.
  • On a graph or blank page draw a large square.
  • Now draw a rectangle on the inside of one of the corners so that the sum of the two sides add to the length of your square. Name the sides a and b.
  • Repeat the process to draw four rectangles altogether, each of side lengths a and b inside your square.
  • Draw in the diagonals of each of the four rectangles in order to construct right-angled triangles in each corner of your large square. This should create a square in the middle of your outside square which is ‘the square on the hypotenuse’ of your corner right-angled triangles. Call this side c.
  • You should end up with a construction looking something like this:

  • Use this construction to prove Pythagoras’ Theorem for a corner triangle of sides a, b and c.

Hint: Shade in a square of area a2 and a square of area b2 and then try to make these equal the square on the hypotenuse, c2.

At this point it is a matter of seeing an area for a2 and b2 and relating this to the area for c2. This can be seen by moving two of the right-angled triangles from the area for a2 + b2 into the area for c2. Note: the tilted square is c2.

To see this puzzle demonstrated you can click here.

More about Pythagoras

The Pythagorean Theorem is one of Euclidean Geometry's most beautiful theorems. It is simple, yet obscure, and is used continuously in mathematics and physics. Evidence of the theorem can be traced far back into Egyptian history with the help of the Rhind Papyrus (1788-1580 BC).

The history of the theorem and its proofs is a fascinating journey, and this activity gives you the opportunity to do some discovering yourself.

A good idea is to search the Internet. But here are some possible starting points. Look out for different proofs of the Pythagorean Theorem and the story of Pythagoras himself.

 

http://staff.hum.ku.dk/dbwagner/Pythagoras/Pythagoras.html

http://www.cut-the-knot.com/pythagoras/

http://www.perseus.tufts.edu/GreekScience/Students/Tim/Pythag'sTheorem.html

http://www.anselm.edu/homepage/dbanach/pyth1.htm

 

 

 

 

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