Right Triangles - The Foundation of Trigonometry
A right triangle is a triangle that contains one angle of exactly 90 degrees. Right triangles are the foundation of trigonometry. Every trigonometric ratio is defined in terms of the sides of a right triangle relative to one of its acute angles.
The relationship a² + b² = c² is named after the Greek philosopher Pythagoras (c. 570–495 BCE), but the underlying pattern was known far earlier: a Babylonian clay tablet called Plimpton 322, dated to around 1800 BCE, lists rows of numbers that form Pythagorean triples over a thousand years before Pythagoras was born. What Pythagoras and his followers contributed was the first known general proof that the relationship holds for every right triangle, not just the specific cases the Babylonians had tabulated – a shift from pattern-spotting to formal mathematical reasoning that shaped the rest of Greek mathematics.
Right triangles remain indispensable in the real world: carpenters use the 3-4-5 triple to check that a corner is truly square without any angle-measuring tools, and surveyors and GPS systems rely on networks of right triangles to calculate distances that cannot be measured directly.
Parts of a Right Triangle
Choose one of the two acute angles – call it θ (theta). The three sides are then named relative to θ:
Hypotenuse – the longest side, always opposite the right angle.
Opposite – the side directly across from angle θ.
Adjacent – the side next to angle θ (not the hypotenuse).
Pythagoras' Theorem
In any right triangle with hypotenuse c and shorter sides a and b:
a² + b² = c²
This lets you find any missing side when two sides are known.
The Angles of a Right Triangle
- One angle is always 90° (the right angle).
- The other two angles are acute (each less than 90°) and they always sum to 90°.
- If one acute angle is θ, the other is 90° − θ (its complement).
Special Right Triangles
| Triangle | Angles | Side ratio |
|---|---|---|
| 45–45–90 | 45°, 45°, 90° | 1 : 1 : √2 |
| 30–60–90 | 30°, 60°, 90° | 1 : √3 : 2 |
Memorising these ratios means you can solve problems involving these angles without a calculator.
Worked Examples
c² = 6² + 8² = 36 + 64 = 100. c = √100 = 10 cm.
b² = 13² − 5² = 169 − 25 = 144. b = √144 = 12 cm.
Ratio 1 : √3 : 2. Multiply each by 5:
Other leg = 5√3 ≈ 8.66 cm. Hypotenuse = 10 cm.
Key Takeaways
- Label sides as hypotenuse, opposite, and adjacent relative to the angle you are working with.
- Pythagoras: a² + b² = c² (c is always the hypotenuse).
- 45–45–90 ratio: 1 : 1 : √2. 30–60–90 ratio: 1 : √3 : 2.
- The two acute angles in a right triangle always sum to 90°.
Practice: Pythagoras' Theorem
Related Topics
Continue exploring related topics: