Pythagoras created one of the most famous mathematical theorems of all time, and it can be used as the subject of a maths project in numerous ways. The basic theory is that in a right angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the remaining two sides. In algebraic terms, this is simply stated as a^2 + b^2 = c^2. This can be proven in numerous ways, and it can also be applied to real world situations like using a ladder.

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## Basic proof using squares

The simplest way to test the Pythagorean Theorem is to use geometric shapes to represent the more complicated operations in the equation. Draw a right-angled triangle with sides of 3cm, 4cm and 5cm on a piece of paper (you can use the same figures in inches, since the theory works regardless of the size of the triangle – this applies for all examples), and carefully cut it out using scissors. Draw three squares based on the length of each side of the triangle, so in the example, one 3cm square, one 4cm square and one 5cm square. You can place the squares beside their related sides of the triangle to test you’ve got the dimensions right. Split each square into smaller, 1cm squares and count them. The 3cm square will have 9 1cm blocks, the 4cm square will have 16 and the final one will have 25. The theorem states that the sum of the 3cm and 4cm squares (9 and 16) will be equal to the 5cm square (25), and some basic addition shows that it is.

## The reverse proof

If any right-angled triangle can be described using the Pythagorean Theorem, it follows logically that the opposite is also true. Using three lengths which can be placed into the theorem (where the sum of two of the squares is equal to the third square), such as 6cm, 8cm and 10cm, try to draw a triangle which doesn’t include a right angle. You’ll find that no matter which numbers you use (as long as they can replace "a," "b" and "c" in the equation), you can only draw a triangle which creates a right angle. This proof was first shown by Euclid (see References 1).

## Rotation proof

Two right-angled triangles can be rotated in a way which shows that Pythagoras was right. Draw two squares side by side, with the left one larger than the right. The length of the sides of the larger square can be called a and the sides of the smaller square can be called length "b." Draw two lines to create two right angled triangles which both have a side equal to "a" and one equal to "b" – this should take the form of two lines joined by a right angle which aren’t parallel to any of the squares’ sides. Cut out the two triangles and place them back into their original positions. Rotate the left triangle 90 degrees anti-clockwise, using the corner where sides "a" and "c" meet as the axis of rotation. Rotate the right triangle 90 degrees clockwise, using the corner where sides "b" and "c" meet as the axis of rotation. You’ll find that the two hypotenuse angles create the outer sides of a square with an area of c^2.

## Is your ladder big enough?

Real-world problems with the Pythagorean Theorem show that it isn’t confined to academic settings. Imagine you have to climb up onto a roof which is 4 m off the ground, but you can’t move your ladder too close because of an uneven patch of ground which extends up to 3 m away from the wall. Work out the length of the ladder you need to reach the roof for a real-world Pythagorean project. You’ll find that you need at least a 5 m ladder to reach the roof. In imperial measurements, you can use values of 8 and 6 feet for the height of the roof and the length of the unusable patch of ground, meaning you’d need a ladder at least 10 feet long.