1:1:√2 ratio

Definition

The 1:1:√2 ratio refers to the specific relationship between the sides of a right triangle, particularly in a 45-45-90 triangle. In this triangle, the two legs are of equal length, while the hypotenuse is √2 times the length of either leg. This relationship arises from the properties of special right triangles and allows for easy calculations involving angles and lengths in geometry.

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 have a length of 'x√2'.
2. The angles in a 45-45-90 triangle are always 45 degrees each for the legs and 90 degrees for the hypotenuse.
3. The 1:1:√2 ratio allows for simple calculations when finding missing sides or angles in problems involving 45-45-90 triangles.
4. This ratio is frequently used in real-life applications such as architecture and design due to its geometric properties.
5. Understanding this ratio is crucial for solving problems related to diagonal measurements in squares and rectangles.

Review Questions

• How can you apply the 1:1:√2 ratio to solve problems involving 45-45-90 triangles?
  • To solve problems with 45-45-90 triangles using the 1:1:√2 ratio, first identify the lengths of the legs. If one leg is known to be 'x', then apply the ratio to find that the other leg is also 'x' and calculate the hypotenuse as 'x√2'. This relationship simplifies finding missing lengths in various geometric scenarios, making it easier to tackle problems that involve this special type of triangle.
• Discuss how the Pythagorean Theorem relates to the 1:1:√2 ratio in a 45-45-90 triangle.
  • The Pythagorean Theorem directly supports the understanding of the 1:1:√2 ratio by confirming that in a 45-45-90 triangle, if both legs are 'x', then according to the theorem $$x^2 + x^2 = c^2$$ results in $$c^2 = 2x^2$$. Taking the square root of both sides shows that $$c = x√2$$, thus reinforcing that the hypotenuse is √2 times longer than each leg and establishing this unique ratio.
• Evaluate how knowledge of the 1:1:√2 ratio can enhance your understanding of trigonometric functions in special right triangles.
  • Knowing the 1:1:√2 ratio significantly enhances understanding of trigonometric functions because it provides a basis for calculating sine, cosine, and tangent values for angles found in special right triangles. For instance, in a 45-degree angle, both sine and cosine equal $$\frac{1}{\sqrt{2}}$$ based on this ratio. Therefore, having this foundational knowledge aids in applying trigonometric ratios effectively when solving more complex problems involving angles and side lengths.

Related terms

45-45-90 Triangle: A special right triangle where the two legs are equal in length, and the angles are 45 degrees, leading to a hypotenuse that is √2 times the length of each leg.

Pythagorean Theorem:

A fundamental principle in geometry stating that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, expressed as $$a^2 + b^2 = c^2$$.

Trigonometric Ratios: Ratios that relate the angles of a triangle to the lengths of its sides, including sine, cosine, and tangent, which can be applied to special right triangles.