45-45-90 Triangle - Algebra-2

1. Fundamental Concepts

Definition: A 45-45-90 triangle is a special right triangle where the two legs are congruent, and each leg forms a 45-degree angle with the hypotenuse.
Properties: The ratio of the lengths of the legs to the hypotenuse is $$\frac{1}{\sqrt{2}}$$ or $$\frac{\sqrt{2}}{2}$$.
Hypotenuse Calculation: If the length of each leg is $$x$$, then the hypotenuse is $$x \cdot \sqrt{2}$$.

2. Key Concepts

Leg-Hypotenuse Relationship: If one leg is $$x$$, the hypotenuse is $$x \cdot \sqrt{2}$$.

Trigonometric Ratios: For a 45-45-90 triangle, $$\sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2}$$.

Application: Used in geometry and trigonometry for simplifying calculations involving angles and lengths.

3. Examples

Example 1 (Basic)

Problem: Find the hypotenuse of a 45-45-90 triangle if each leg measures 5 units.

Step-by-Step Solution:

Given that each leg is 5 units, use the formula for the hypotenuse: $$5 \cdot \sqrt{2}$$.
The hypotenuse is approximately $$5 \cdot 1.414 = 7.07$$ units.

Validation: Substitute into the Pythagorean theorem: $$5^2 + 5^2 = (\text{{hypotenuse}})^2$$. Simplified: $$25 + 25 = 50$$; Hypotenuse: $$\sqrt{50} = 5\sqrt{2} \approx 7.07$$ ✓

Example 2 (Intermediate)

Problem: If the hypotenuse of a 45-45-90 triangle is 10 units, find the length of each leg.

Step-by-Step Solution:

Given the hypotenuse is 10 units, use the relationship: $$x \cdot \sqrt{2} = 10$$.
Solve for $$x$$: $$x = \frac{10}{\sqrt{2}} = \frac{10 \cdot \sqrt{2}}{2} = 5\sqrt{2}$$.

Validation: Substitute back into the relationship: $$5\sqrt{2} \cdot \sqrt{2} = 10$$; Simplified: $$5 \cdot 2 = 10$$ ✓

4. Problem-Solving Techniques

Visual Strategy: Draw the triangle and label all sides and angles clearly.
Error-Proofing: Always check your solution using the Pythagorean theorem.
Concept Reinforcement: Practice problems with different side lengths to reinforce understanding.