This is a fundamental theorem in geometry that you will use frequently in many other topics, including Trigonometry, Surface Area, and Volume. It applies only to right-angled triangles.

1. The Core Concepts

The Theorem

"In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides."

• Hypotenuse (c): This is the most important side. It is always the longest side and is always opposite the right angle (90°).

• The other two sides (a, b): These are the two sides that form the right angle.

The Formula:

a² + b² = c²

The Converse of the Theorem

This is used to prove that a triangle is right-angled.

"If the lengths of the sides of a triangle satisfy the relationship a² + b² = c² (where c is the longest side), then the angle opposite the longest side is a right angle."

• How to use it: Take the lengths of the three sides. Square all of them. Check if the sum of the two smaller squares equals the largest square. If it does, the triangle is right-angled.

2. Pythagorean Triples (Important Shortcut!)

Pythagorean triples are sets of three positive integers that satisfy Pythagoras' theorem. Memorizing the common ones can save you a lot of time in exams.

• Common Triples:

  • (3, 4, 5)

  • (5, 12, 13)

  • (8, 15, 17)

  • (7, 24, 25)

• Multiples also work: Any multiple of a Pythagorean triple is also a triple.

  • Multiple of (3, 4, 5) by 2 gives (6, 8, 10).

  • Multiple of (5, 12, 13) by 3 gives (15, 36, 39).

Exam Tip: If you see a right-angled triangle with two sides given as 8 cm and 10 cm, you can immediately recognize it's a multiple of (?, 4, 5) by 2. The missing side must be 6 cm (since 3x2=6). This is much faster than calculating √(10² - 8²).

How to Solve Typical Exam Problems

Type 1: Finding an Unknown Side

1. Identify the hypotenuse (c). It is the side opposite the 90° angle.

2. To find the hypotenuse (c): Use addition. c = √(a² + b²).

3. To find a shorter side (a or b): Use subtraction. a = √(c² - b²).

Type 2: Proving Riders

This is a very common type of question. The strategy is to find the right-angled triangles in the diagram and use them to build equations.

1. Identify all the right-angled triangles in the figure.

2. Apply Pythagoras' theorem to each of these triangles and write down the resulting equations.

   • In ΔABX, AB² = AX² + BX² --- (1)

   • In ΔACX, AC² = AX² + CX² --- (2)

3. Manipulate the equations (e.g., subtract (2) from (1)) to eliminate common terms (like AX²) and arrive at the expression you need to prove.

Common Pitfalls & Exam Strategy

• Pitfall 1: Wrongly Identifying the Hypotenuse. This is the biggest mistake. c is NOT always the unknown side. It is ALWAYS the side opposite the right angle.

• Pitfall 2: Mixing up Addition and Subtraction. Remember: Hypotenuse² = Side² + Side². You only use addition when finding the hypotenuse. For any other side, you must subtract.

• Exam Strategy: When a question involves a shape like a rhombus, remember its properties. The diagonals of a rhombus bisect each other at 90 degrees, creating four right-angled triangles for you to work with.

• Answering: Unless the question asks for a decimal value, it is often best to leave answers in surd form (e.g., √32 or simplified to 4√2). This is more accurate.