1. Pythagoras' Theorem
What is Pythagoras' Theorem?
Pythagoras' theorem is a formula that connects the three sides of any right-angled triangle. It only works for triangles that have a 90-degree angle (a right angle).
The three sides:
- Hypotenuse: This is the longest side of a right-angled triangle. It is always the side directly opposite the right angle.
- Two shorter sides: These are the other two sides that form the right angle.
The formula:
a² + b² = c²
Where:
- c = length of the hypotenuse (longest side)
- a and b = lengths of the two shorter sides
It doesn't matter which shorter side you call a and which you call b.
Finding the hypotenuse:
If you know the two shorter sides and need to find the hypotenuse, follow these steps:
- Square both shorter sides (multiply each by itself)
- Add these two numbers together
- Take the positive square root of the answer
Formula: c = √(a² + b²)
Example: A right-angled triangle has shorter sides of 6 cm and 8 cm. Find the hypotenuse.
- c = √(6² + 8²)
- c = √(36 + 64)
- c = √100
- c = 10 cm
Finding a shorter side:
If you know the hypotenuse and one shorter side, and need to find the other shorter side:
- Square the hypotenuse and the known shorter side
- Subtract the smaller value from the larger value
- Take the positive square root of the answer
Formula: a = √(c² - b²)
Example: A right-angled triangle has a hypotenuse of 13 cm and one shorter side of 5 cm. Find the other shorter side.
- a = √(13² - 5²)
- a = √(169 - 25)
- a = √144
- a = 12 cm
Important tips:
- The hypotenuse must always be the longest side. If your answer makes it shorter than another side, you've made a mistake.
- When finding a shorter side, always subtract before taking the square root. You add when finding the hypotenuse.
- For multi-step problems, don't round your answers until the very end to keep them accurate.