45-45-90 Triangle

Definition

A 45-45-90 triangle is a right triangle in which all three angles are in the ratio of 45 degrees, 45 degrees, and 90 degrees. This unique triangle has special properties that make it particularly useful in various mathematical and geometric applications.

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 third angle is 90 degrees (a right angle).
2. The two equal sides of a 45-45-90 triangle are both the same length, and the hypotenuse is $\sqrt{2}$ times the length of the equal sides.
3. The trigonometric ratios (sine, cosine, and tangent) of a 45-45-90 triangle have special, easily remembered values.
4. 45-45-90 triangles are often used in engineering, architecture, and other fields where precise geometric relationships are important.
5. Verifying trigonometric identities and simplifying trigonometric expressions often involve the special properties of 45-45-90 triangles.

Review Questions

• Explain how the properties of a 45-45-90 triangle can be used to verify trigonometric identities.
  • The special angles and side lengths of a 45-45-90 triangle can be used to verify trigonometric identities, such as $\sin(45^{\circ}) = \cos(45^{\circ})$ or $\tan(45^{\circ}) = 1$. By substituting the known trigonometric ratios for a 45-45-90 triangle into an identity, you can quickly determine if the identity holds true, making it a useful tool for verifying and simplifying trigonometric expressions.
• Describe how the Pythagorean Theorem relates to the side lengths of a 45-45-90 triangle.
  • In a 45-45-90 triangle, the Pythagorean Theorem can be used to determine the relationship between the lengths of the two equal sides and the hypotenuse. Specifically, if the length of one of the equal sides is $a$, then the length of the hypotenuse is $\sqrt{2}a$. This is because the square of the hypotenuse is equal to the sum of the squares of the other two sides, which are equal in a 45-45-90 triangle.
• Analyze how the properties of a 45-45-90 triangle can be applied in real-world engineering and architectural design problems.
  • The special characteristics of a 45-45-90 triangle, such as the equal side lengths and the $\sqrt{2}$ ratio between the hypotenuse and the equal sides, make it a valuable tool in engineering and architectural design. Engineers and architects can use 45-45-90 triangles to create structurally sound and aesthetically pleasing designs, such as in the construction of roofs, bridges, and other structures that require precise geometric relationships. Additionally, the ease of calculating trigonometric ratios for a 45-45-90 triangle allows for efficient design calculations and analysis.

Related terms

Right Triangle: A right triangle is a triangle in which one of the angles is a 90-degree angle, or a right angle.

Isosceles Triangle: An isosceles triangle is a triangle in which two sides are equal in length, and the angles opposite those equal sides are also equal.

Pythagorean Theorem:

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.