45-45-90 triangle

Definition

A 45-45-90 triangle is an isosceles right triangle where the angles are 45 degrees, 45 degrees, and 90 degrees. This specific type of triangle has unique properties that relate the lengths of its sides, making it significant in the study of trigonometric functions and identities. Understanding this triangle helps to simplify calculations involving angles and distances, especially in geometric contexts.

5 Must Know Facts For Your Next Test

1. In a 45-45-90 triangle, the lengths of the legs are equal, and if one leg is of length 'x', the hypotenuse will be 'x√2'.
2. The angles in a 45-45-90 triangle directly relate to the sine and cosine values: both sine(45°) and cosine(45°) equal $$\frac{\sqrt{2}}{2}$$.
3. This triangle provides a quick way to find distances when dealing with diagonal lines in grid-based problems.
4. The area of a 45-45-90 triangle can be calculated using the formula $$\text{Area} = \frac{1}{2} x^2$$ where 'x' is the length of each leg.
5. It is often used in real-world applications such as architecture and engineering due to its simplicity in calculations involving angles.

Review Questions

• How do the properties of a 45-45-90 triangle simplify trigonometric calculations?
  • The properties of a 45-45-90 triangle simplify trigonometric calculations because both legs are of equal length. This means that when finding sine and cosine values for 45 degrees, they can be directly represented as $$\frac{\sqrt{2}}{2}$$ without additional calculations. Additionally, knowing that the hypotenuse is always $$x√2$$ allows for quick determination of distances in problems involving these triangles.
• Demonstrate how to derive the lengths of sides in a 45-45-90 triangle using the Pythagorean Theorem.
  • To derive the lengths of sides in a 45-45-90 triangle using the Pythagorean Theorem, let each leg be 'x'. According to the theorem, we have $$a^2 + b^2 = c^2$$. Here, both legs 'a' and 'b' are equal (both are 'x'), thus we have $$x^2 + x^2 = c^2$$ or $$2x^2 = c^2$$. Solving for 'c' gives us $$c = x√2$$, showing how side lengths relate to one another in this type of triangle.
• Evaluate how understanding a 45-45-90 triangle can enhance problem-solving skills in real-world scenarios.
  • Understanding a 45-45-90 triangle enhances problem-solving skills in real-world scenarios by allowing individuals to quickly compute distances and angles. For instance, in construction, knowing how to apply these triangles can help determine roof slopes or wall heights with minimal effort. This knowledge enables efficient calculations without needing complex tools or methods, thus improving productivity and accuracy in practical applications.

Related terms

Isosceles Triangle: A triangle that has at least two sides of equal length and two angles that are equal.

Pythagorean Theorem: A fundamental relation in Euclidean geometry among the three sides of a right triangle, stating that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Trigonometric Ratios: Ratios of the lengths of sides in a right triangle, commonly used to define the sine, cosine, and tangent functions.