30-60-90 Triangle - Algebra-2

1. Fundamental Concepts

Definition: A 30-60-90 triangle is a special right triangle where the angles are 30°, 60°, and 90°. The sides opposite these angles are in a fixed ratio of $$1 : \sqrt{3} : 2$$.
Properties: In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is $$\sqrt{3}$$ times the length of the shorter leg.
Applications: These triangles are useful in trigonometry and geometry for solving problems involving angles and side lengths.

2. Key Concepts

Side Ratio: $$\text{{Shorter Leg}} : \text{{Longer Leg}} : \text{{Hypotenuse}} = 1 : \sqrt{3} : 2$$

Trigonometric Ratios: For a 30-60-90 triangle, \sin(30^\circ) = \frac{1}{2}, \cos(30^\circ) = \frac{\sqrt{3}}{2}, \tan(30^\circ) = \frac{1}{\sqrt{3}}$$

Application: Used to solve real-world problems involving angles and distances, such as calculating heights or distances in construction and navigation.

3. Examples

Example 1 (Basic)

Problem: Find the length of the hypotenuse if the shorter leg of a 30-60-90 triangle is 4 units.

Step-by-Step Solution:

The hypotenuse is twice the length of the shorter leg: $$\text{{Hypotenuse}} = 2 \cdot 4 = 8$$

Validation: Given the shorter leg is 4, the hypotenuse should be 8. ✓

Example 2 (Intermediate)

Problem: If the hypotenuse of a 30-60-90 triangle is 10 units, find the lengths of the other two sides.

Step-by-Step Solution:

The shorter leg is half the hypotenuse: $$\text{{Shorter Leg}} = \frac{10}{2} = 5$$
The longer leg is $$\sqrt{3}$$ times the shorter leg: $$\text{{Longer Leg}} = 5 \cdot \sqrt{3} = 5\sqrt{3}$$

Validation: Given the hypotenuse is 10, the shorter leg is 5 and the longer leg is $$5\sqrt{3}$$. ✓

4. Problem-Solving Techniques

Visual Strategy: Draw the triangle and label all known sides and angles.
Error-Proofing: Always check that the side ratios match the standard 1 : $$\sqrt{3}$$ : 2 pattern.
Concept Reinforcement: Practice with various problems to reinforce understanding of the relationships between the sides and angles.