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Pythagoras Theorem

Pythagoras theorem deals with the relationship between the three sides of a right-angled triangle. The longest side in a right-angled triangle is called the hypotenuse.

Pythagoras theorem

 

Pythagoras Theorem states that in any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

This can be written as : c2 = a2 + b2

 

 

Pythagorean triad

A Pythagorean triad is a set of three positive integers (whole numbers) – a, b and c – such that c2 = a2 + b2. For example; 3, 4 and 5 is a Pythagoras triad because 52 = 32 + 42   (25 = 9 + 16).

This triad was know to the ancient Egyptians and Babylonians – they found that a triangle that had sides of 3, 4 and 5 units in length was always a right-angled triangle. But they were unable to explain why these numbers always gave a right angle; although this did not stop them from using this concept while building pyramids and other projects.

However, it was a religious group led by the famous Greek philosopher/mathematician Pythagoras, around 550 BC, that discovered the relationship between the sides of all right-angled triangles. Hence this theorem is named after Pythagoras.

 

Pythagoras Theorem can also be illustrated by the following geometric method. Let us take a right-angled triangle with sides 3, 4 and 5 units, and construct squares (with unit dimensions) on each side of the triangle as shown in the diagram.

Pythagoras theorem_illustration

 

We have seen that for all right-angled triangles,  c2 = a2 + b2

Since the areas of the squares constructed on the sides of 32, 42 and 52 respectively, the theorem can be restated as:

The square on the hypotenuse is equal to the sum of the squares on the two smaller sides.

Looking at the three squares, we can see that area of the square with side 3 is 9 square units, and that with side 4 is 16 square units and the square with side 5 is 25 square units. Adding 9 and 16 gives 25 which is the area of the largest square. This proves the Pythagoras Theorem.

 

Example 1: Calculate the length of the hypotenuse in the triangle below.

Applying Pythagoras Theorem for this right-angled triangle,  Pythagoras theorem_example1

52  +  122  =  x2

x2  =  25  +  144

x  =  sqrt{169}

x  =  13 cm

 

Example 2: What is the length of the hypotenuse in this triangle?

x2  =  32  +  52  Pythagoras theorem_example2

x2  =  9 + 25

x  =  sqrt{34}

x  =  5.8 cm (correct to 1 decimal place)

 

Learn more about Pythagoras Theorem here –

Examples of Pythagoras Theorem

Converse of Pythagoras Theorem

Finding a short side

Advanced problems using Pythagoras Theorem