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.
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 Questions
- Find the hypotenuse of a right triangle with legs 9 cm and 12 cm.
- A right triangle has hypotenuse 25 cm and one leg 7 cm. Find the missing leg.
- In a 45–45–90 triangle the hypotenuse is 10 cm. Find the legs.
- In a 30–60–90 triangle the hypotenuse is 14 cm. Find both legs.
- A ladder 5 m long leans against a wall. Its foot is 3 m from the wall. How high up the wall does it reach?