45-45-90 triangle

Definition

A 45-45-90 triangle is a specific type of right triangle where the two non-hypotenuse angles are each 45 degrees. This unique configuration leads to a distinct relationship between the lengths of its sides, which is essential in understanding right triangle trigonometry. The side lengths of a 45-45-90 triangle follow the ratio of $1 : 1 : \sqrt{2}$, making it easier to solve for missing side lengths and apply in various mathematical contexts.

5 Must Know Facts For Your Next Test

1. In a 45-45-90 triangle, both legs are congruent, meaning they have equal lengths.
2. If each leg of a 45-45-90 triangle measures 'x', then the hypotenuse will measure 'x\sqrt{2}'.
3. The angles in a 45-45-90 triangle add up to 180 degrees, with one angle being 90 degrees and the other two being 45 degrees each.
4. This type of triangle can be derived from cutting an isosceles right triangle in half along its height.
5. These triangles are often used in various applications, including architecture and design, due to their consistent side ratios.

Review Questions

• How does the side length ratio in a 45-45-90 triangle facilitate problem-solving in trigonometry?
  • The side length ratio of $1 : 1 : \sqrt{2}$ in a 45-45-90 triangle simplifies calculations when determining unknown side lengths. Because both legs are equal, it allows for quick solutions when one leg's length is known; the hypotenuse can be easily found by multiplying that leg's length by $\sqrt{2}$. This property makes it especially useful in problems involving right triangles.
• Compare and contrast the properties of a 45-45-90 triangle with those of other special triangles, like a 30-60-90 triangle.
  • While both 45-45-90 and 30-60-90 triangles have specific angle measures that lead to consistent side ratios, they differ in their properties. A 30-60-90 triangle has sides in the ratio of $1 : \sqrt{3} : 2$, while a 45-45-90 triangle has equal legs and a hypotenuse proportional to $\sqrt{2}$. These distinctions influence their applications; for instance, 30-60-90 triangles are commonly seen in geometric problems involving circles and hexagons.
• Evaluate how understanding the properties of a 45-45-90 triangle contributes to broader applications in mathematics and real-world scenarios.
  • Understanding a 45-45-90 triangle is crucial for applying geometric principles across various fields such as engineering, physics, and architecture. For example, this knowledge allows architects to design structures that incorporate right angles efficiently. Additionally, it aids in creating accurate models or designs where symmetry is essential. Mastery of these triangles lays foundational skills that enhance problem-solving capabilities in more complex mathematical challenges.

Related terms

Hypotenuse:

The longest side of a right triangle, opposite the right angle, which can be calculated using the Pythagorean theorem.

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: Relationships between the angles and sides of a triangle that allow for calculations involving sine, cosine, and tangent.