Pythagoras' Theorem

What is Pythagoras' Theorem?

Pythagoras' theorem is a mathematical rule that works for right-angled triangles only. A right-angled triangle is a triangle that has one angle of exactly 90 degrees (a square corner, like the corner of a book).

The theorem tells us how the three sides of a right-angled triangle are related to each other. It was discovered by a Greek mathematician called Pythagoras over 2500 years ago, and it is one of the most useful formulas in mathematics.

The Three Sides of a Right-Angled Triangle

Before we learn the theorem, we need to understand the three sides:

1. The hypotenuse (say: hy-POT-en-use)

• This is the longest side of a right-angled triangle
• It is always the side opposite the right angle (across from it)
• It never touches the right angle corner

2. The two shorter sides

• These are the two sides that form the right angle
• They meet at the square corner
• We can call them any letters (often a and b)

Here's a simple diagram to show you:

        c (hypotenuse - longest side)
       /|
      / |
     /  | b (shorter side)
    /   |
   /____| 
   a     
(shorter side)

The right angle is where sides a and b meet. The hypotenuse (c) is opposite this corner.

The Pythagoras' Theorem Formula

The theorem states:

a² + b² = c²

Where:

• c is the length of the hypotenuse (the longest side)
• a and b are the lengths of the two shorter sides
• It doesn't matter which shorter side you call a and which you call b

What does this mean in plain English?

If you take the length of one shorter side and square it (multiply it by itself), then take the length of the other shorter side and square it, and add these two numbers together, you get the same answer as squaring the hypotenuse.

Example: If the two shorter sides are 3 cm and 4 cm, and the hypotenuse is 5 cm:

• 3² = 3 × 3 = 9
• 4² = 4 × 4 = 16
• 5² = 5 × 5 = 25
• Check: 9 + 16 = 25 ✓ (This proves it's a right-angled triangle!)

Using Pythagoras' Theorem to Find the Hypotenuse

If you know the lengths of the two shorter sides, you can find the length of the hypotenuse.

Step-by-step method:

Step 1: Square each of the two shorter sides (multiply each number by itself)

Step 2: Add these two squared numbers together

Step 3: Take the square root of your answer (this is the reverse of squaring)

Formula: c = √(a² + b²)

Notice that when finding the hypotenuse, you add the squared numbers before taking the square root.

Example: Find the hypotenuse if the two shorter sides are 5 cm and 12 cm.

Step 1: Square both sides

• 5² = 25
• 12² = 144

Step 2: Add them together

• 25 + 144 = 169

Step 3: Take the square root

• √169 = 13

Answer: The hypotenuse is 13 cm