The Right-Angle Secret: Pythagoras' Theorem

Ancient Egyptian builders are said to have used a clever rope trick to make perfectly square corners. They took a rope with 12 equally spaced knots and formed a triangle with sides of 3, 4, and 5 gaps. The corner between the shortest sides was always a perfect right angle. How could a simple triangle guarantee a right angle? That is the mystery you are about to unlock.

A right-angled triangle has one angle that is exactly 90°, shown by a small square in the corner. The three sides have special names. The two shorter sides — the ones that form the right angle — are called the legs. We label them a and b. The longest side, which is always directly opposite the right angle, is called the hypotenuse. We label it c. Pythagoras' theorem states: a² + b² = c² The notation a² means 'a squared', which is a × a. So the theorem says: add the square of each leg together, and you get the square of the hypotenuse. This is true for every right-angled triangle, no matter how big or small. There are two types of problem you will meet: 1. Finding the hypotenuse (c): Use a² + b² = c², then take the square root of both sides to find c. Square root (√) means: 'which number, multiplied by itself, gives this result?' √25 = 5 because 5 × 5 = 25. 2. Finding a missing leg (a or b): Rearrange the formula. If you want a: a² = c² − b², then find √(c² − b²). WORKED EXAMPLE 1 — Finding the hypotenuse: A right-angled triangle has legs a = 3 and b = 4. Find c. Step 1: a² + b² = 3² + 4² = 9 + 16 = 25 Step 2: c² = 25, so c = √25 = 5 Answer: the hypotenuse is 5. WORKED EXAMPLE 2 — Finding a missing leg: A right-angled triangle has hypotenuse c = 13 and one leg b = 5. Find a. Step 1: a² = c² − b² = 13² − 5² = 169 − 25 = 144 Step 2: a = √144 = 12 Answer: the missing leg is 12. Notice that in both examples you could work out the square roots mentally because the answers were perfect squares (25 and 144). In harder problems you may need a calculator, but the steps are identical.