45-45-90 triangle

Definition

A 45-45-90 triangle is a special type of right triangle where the two non-hypotenuse angles are each 45 degrees. This unique triangle has side lengths that follow a specific ratio: the lengths of the legs are equal, and the length of the hypotenuse is equal to the length of a leg multiplied by the square root of 2. Understanding this triangle helps in solving problems related to right triangle trigonometry, making it easier to determine unknown side lengths and angles in various applications.

5 Must Know Facts For Your Next Test

1. In a 45-45-90 triangle, if each leg has a length of 'x', then the hypotenuse will always be 'x√2'.
2. The angles in a 45-45-90 triangle always sum up to 180 degrees, with one angle being 90 degrees and the other two being 45 degrees each.
3. This type of triangle is often encountered in real-life scenarios, such as calculating distances or heights in architecture and design.
4. When converting between degrees and radians, 45 degrees corresponds to π/4 radians, which can be important in more advanced trigonometric calculations.
5. The properties of a 45-45-90 triangle make it useful for simplifying calculations in both geometry and trigonometry.

Review Questions

• How do you derive the relationship between the legs and hypotenuse in a 45-45-90 triangle using the Pythagorean Theorem?
  • To derive the relationship in a 45-45-90 triangle, you start with two legs of equal length, let's say 'x'. According to the Pythagorean Theorem, we have $$a^2 + b^2 = c^2$$. Substituting 'x' for both legs gives us $$x^2 + x^2 = c^2$$ or $$2x^2 = c^2$$. Taking the square root results in $$c = x√2$$, showing how the hypotenuse relates to the legs.
• Explain how the properties of a 45-45-90 triangle can be applied to solve real-world problems involving heights or distances.
  • The properties of a 45-45-90 triangle can be applied in real-world situations where you need to calculate heights or distances. For instance, if you know that a structure has angles of elevation forming a 45-degree angle with the ground, you can use the ratio of the legs and hypotenuse to easily find unknown dimensions. This makes it practical for architects or engineers when designing structures that rely on equal leg lengths for support.
• Evaluate how understanding 45-45-90 triangles enhances your problem-solving skills in trigonometry and geometry applications.
  • Understanding 45-45-90 triangles significantly enhances problem-solving skills as it provides a clear and straightforward way to tackle various geometric and trigonometric problems. By recognizing this specific type of triangle, you can quickly apply its properties to find missing lengths or angles without complex calculations. This not only saves time but also builds confidence when approaching more complicated problems that may involve these foundational concepts.

Related terms

Isosceles Triangle: A triangle that has at least two sides of equal length, which is a defining feature of the 45-45-90 triangle.

Pythagorean Theorem: A fundamental principle in geometry stating that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Trigonometric Ratios: The ratios of the lengths of sides in a right triangle, specifically sine, cosine, and tangent, which can be applied in understanding and solving problems involving 45-45-90 triangles.