Pythagoras' Theorem

Pythagoras who?

• Pythagoras was an ancient Greek mathematician who lived over 2500 years ago. 
• He is best known for discovering the Pythagorean Theorem, which is one of the fundamental principles in geometry, used for finding missing side lengths in right-angled triangles.

 
Pythagoras' Theorem Explained

Pythagoras' Theorem states that in any right-angled triangle

• The square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides:

$$a^2 + b^2 = c^2$$

 

 

where:

• c is the hypotenuse (the side opposite the right angle and the longest side)
• a and b are the two shorter sides

Key Note: It does not matter which of the two shorter sides is labeled as a or b, but c must always be the hypotenuse

 

Finding the Hypotenuse

If you need to find the hypotenuse (c):

1. Square the lengths of both shorter sides.
2. Add these squared values.
3. Take the square root to find c.

$$c = \sqrt{a^2 + b^2}$$

 

Example: Finding the Hypotenuse

A right-angled triangle has sides of 6 cm and 8 cm. Find the hypotenuse.

Final Answer: The hypotenuse is 10 cm.

 

Finding a Shorter Side

If you need to find one of the shorter sides:

1. Square the lengths of the hypotenuse and the known shorter side.
2. Subtract these squared values.
3. Take the square root to find the missing side.

$$a = \sqrt{c^2 - b^2}$$

 

Example: Finding a Shorter Side

A right-angled triangle has a hypotenuse of 13 cm and one shorter side of 5 cm. Find the missing side.

$$a = \sqrt{13^2 - 5^2} \quad  a = \sqrt{169 - 25}  \\  a = \sqrt{144} \quad  a = 12$$

Final Answer: The missing side is 12 cm.