The Statement of Pythagoras' Theorem
Pythagoras' theorem applies exclusively to right-angled triangles, which have one angle measuring exactly 90 degrees. The side opposite this right angle is called the hypotenuse and is always the longest side.
The theorem states that the square of the hypotenuse's length equals the sum of the squares of the other two sides' lengths. Using symbols, if the hypotenuse is c and the other sides are a and b, then: a² + b² = c².
This relationship allows us to connect the three side lengths mathematically and is the foundation for many geometric calculations involving right triangles.
- Only applies to right-angled triangles.
- Hypotenuse is the side opposite the right angle and is the longest side.
- Formula: a² + b² = c², where c is the hypotenuse.
Using Pythagoras' Theorem to Find a Missing Side
If two side lengths of a right triangle are known, the third can be found by rearranging the formula.
When the hypotenuse length is unknown, calculate it by taking the square root of the sum of the squares of the other two sides.
When one of the shorter sides is unknown, subtract the square of the known shorter side from the square of the hypotenuse, then take the square root to find the missing side.
- To find hypotenuse: c = √(a² + b²)
- To find a shorter side: a = √(c² - b²) or b = √(c² - a²)
Worked Example 1: Finding the Hypotenuse
Consider a right triangle with shorter sides of lengths 3 cm and 4 cm. To find the hypotenuse c:
Calculate c² = 3² + 4² = 9 + 16 = 25. Then c = √25 = 5 cm.
Therefore, the hypotenuse is 5 cm long. This confirms the classic 3-4-5 right triangle.
- Given: a = 3 cm, b = 4 cm
- Calculate c² = 3² + 4² = 25
- Find c = √25 = 5 cm
Worked Example 2: Finding a Shorter Side
Suppose a right triangle has a hypotenuse of 13 m and one shorter side of 5 m. To find the other shorter side b:
Calculate b² = 13² - 5² = 169 - 25 = 144. Then b = √144 = 12 m.
Hence, the missing side length is 12 m. This is another common Pythagorean triple: 5-12-13.
- Given: c = 13 m, a = 5 m
- Calculate b² = 169 - 25 = 144
- Find b = √144 = 12 m
Practical Tips for Applying Pythagoras' Theorem
Always identify the right angle first to confirm the triangle is right-angled before applying the theorem.
Label the sides clearly, marking the hypotenuse as c to avoid confusion.
Use a calculator to find square roots accurately and check your working by squaring the answer to verify it fits the formula.
- Confirm the triangle is right-angled.
- Label sides with a, b (shorter sides) and c (hypotenuse).
- Check answers by substitution back into the formula.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Identify the triangle type | Confirm the triangle has a right angle | Triangle with one 90° angle | Marks awarded for correct identification |
| Label sides | Assign a, b to shorter sides and c to hypotenuse | Sides labelled a=3, b=4, c=? | Clear labelling avoids errors and earns method marks |
| Apply formula | Use a² + b² = c² | Calculate c² = 3² + 4² = 25 | Correct formula application is essential for marks |
| Calculate missing side | Find square root or subtract squares as needed | c = √25 = 5 | Accurate calculation and units required |
| Check result | Verify by substituting back | Check 3² + 4² = 5² | Demonstrates understanding and can gain method marks |
Active recall
Quick check
1. What is the hypotenuse in a right-angled triangle?
The hypotenuse is the side opposite the right angle and is the longest side of the triangle.
2. If a right triangle has sides 6 cm and 8 cm, how do you find the hypotenuse?
Use Pythagoras' theorem: c = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
3. How do you find a shorter side if the hypotenuse is 10 m and the other side is 6 m?
Calculate the missing side as √(10² - 6²) = √(100 - 36) = √64 = 8 m.
4. Can Pythagoras' theorem be used on any triangle?
No, it only applies to right-angled triangles.