45-45-90 Triangle

Definition

A 45-45-90 triangle is a right triangle with two equal angles of 45 degrees and one angle of 90 degrees. This type of triangle has unique properties that make it useful in various mathematical and real-world applications, particularly in the context of right triangle trigonometry.

5 Must Know Facts For Your Next Test

1. In a 45-45-90 triangle, the two equal angles are both 45 degrees, and the hypotenuse is $\sqrt{2}$ times the length of either of the equal sides.
2. The sides of a 45-45-90 triangle are in the ratio of 1:1:$\sqrt{2}$, meaning the two equal sides are of the same length, and the hypotenuse is $\sqrt{2}$ times the length of the equal sides.
3. The trigonometric ratios for a 45-45-90 triangle are: $\sin(45^{\circ}) = \cos(45^{\circ}) = \frac{1}{\sqrt{2}}$, and $\tan(45^{\circ}) = 1$.
4. 45-45-90 triangles are often used in engineering, architecture, and design, as they provide a simple and symmetrical way to construct and analyze structures.
5. The properties of a 45-45-90 triangle can be used to solve a variety of problems in right triangle trigonometry, such as finding missing side lengths or angle measures.

Review Questions

• Explain the relationship between the sides of a 45-45-90 triangle and how this relates to the trigonometric ratios.
  • In a 45-45-90 triangle, the two equal sides are of the same length, and the hypotenuse is $\sqrt{2}$ times the length of the equal sides. This means that the sides of the triangle are in the ratio of 1:1:$\sqrt{2}$. This unique property of 45-45-90 triangles leads to simplified trigonometric ratios, where $\sin(45^{\circ}) = \cos(45^{\circ}) = \frac{1}{\sqrt{2}}$, and $\tan(45^{\circ}) = 1$. These straightforward trigonometric ratios make 45-45-90 triangles particularly useful in a variety of mathematical and real-world applications.
• Describe how the properties of a 45-45-90 triangle can be applied to solve problems in right triangle trigonometry.
  • The unique properties of a 45-45-90 triangle, such as the 1:1:$\sqrt{2}$ side ratio and the simplified trigonometric ratios, can be leveraged to solve a variety of problems in right triangle trigonometry. For example, if you know the length of one side of a 45-45-90 triangle, you can use the side ratio to easily calculate the lengths of the other sides. Additionally, the simplified trigonometric ratios can be used to quickly determine missing angle measures or side lengths, making 45-45-90 triangles a valuable tool in solving right triangle problems.
• Analyze the practical applications of 45-45-90 triangles in fields such as engineering, architecture, and design, and explain how their properties contribute to these applications.
  • 45-45-90 triangles are widely used in engineering, architecture, and design due to their unique properties and symmetry. The 1:1:$\sqrt{2}$ side ratio and the simplified trigonometric ratios make 45-45-90 triangles easy to construct and analyze, allowing for efficient and precise design and construction of structures. For example, in engineering, 45-45-90 triangles are often used in the design of trusses, beams, and other load-bearing components due to their inherent stability and strength. In architecture, 45-45-90 triangles are commonly used in the design of roofs, staircases, and other building elements to create visually appealing and structurally sound designs. Additionally, the symmetry and simplicity of 45-45-90 triangles make them valuable in a variety of design applications, such as in product design, graphic design, and even urban planning.

Related terms

Right Triangle: A triangle with one 90-degree angle, also known as a right angle.

Isosceles Triangle: A triangle with two equal sides, and consequently, two equal angles.

Trigonometric Ratios: The ratios of the sides of a right triangle, such as sine, cosine, and tangent, which are used to solve problems involving angles and side lengths.