1. What Are Special Right Triangles?
Special right triangles are right-angled triangles with angle measures that create fixed and predictable side ratios. These triangles help you find missing side lengths quickly without using the Pythagoras Theorem every time.
The two important special right triangles are:
- 30-60-90 triangle
- 45-45-90 triangle
2. 30-60-90 Triangle
A 30-60-90 triangle is a right triangle with angles 30°, 60°, and 90°.
This triangle always has the same side ratio:
\(1 : \sqrt{3} : 2\)
(60°)
A
/ \
/ \
/ \
(30°)B-------C (90°)
2.1. Side Ratios
Let the shortest side (opposite 30°) be \(x\). Then:
- Side opposite 30° → \(x\)
- Side opposite 60° → \(x\sqrt{3}\)
- Hypotenuse (opposite 90°) → \(2x\)
These ratios always stay the same no matter the size of the triangle.
2.2. Quick Reasoning
A 30° angle is created by cutting an equilateral triangle in half. So the hypotenuse is always twice the shortest side.
2.3. Example
If the shortest side is \(5\):
- Side opposite 60° = \(5\sqrt{3}\)
- Hypotenuse = \(10\)
3. 45-45-90 Triangle
A 45-45-90 triangle is an isosceles right triangle with two angles of 45° and a right angle of 90°.
This triangle always has the side ratio:
\(1 : 1 : \sqrt{2}\)
A (45°)
|\
| \
| \
(90°) B----C (45°)
3.1. Side Ratios
Let each equal leg be \(x\). Then:
- Leg 1 → \(x\)
- Leg 2 → \(x\)
- Hypotenuse → \(x\sqrt{2}\)
The hypotenuse is always \(\sqrt{2}\) times either leg.
3.2. Quick Reasoning
A 45-45-90 triangle is formed by cutting a square diagonally. This is why the two legs are equal and the diagonal becomes \(x\sqrt{2}\).
3.3. Example
If each leg is \(6\):
Hypotenuse = \(6\sqrt{2}\)
4. Why Special Right Triangles Are Useful
Special right triangles allow fast calculation of missing sides without doing long calculations.
They are widely used in:
- Trigonometry basics
- Geometry problems involving heights/distances
- Right-triangle proofs
- Construction and architecture
- Coordinate geometry