The 90-90-45 triangle is a special type of right triangle that exhibits unique properties, making it a fundamental concept in geometry and trigonometry. This triangle is characterized by its three angles, which measure 90, 90, and 45 degrees, respectively. The 90-90-45 triangle is also known as an isosceles right triangle, since two of its sides are equal in length. In this article, we will delve into the properties of the 90-90-45 triangle, exploring its geometric and trigonometric characteristics, and discussing its practical applications.
Key Points
- The 90-90-45 triangle is an isosceles right triangle with two equal sides and two 90-degree angles.
- The triangle's angles are in the ratio 1:1:√2, which is a fundamental property of 45-45-90 triangles.
- The side lengths of a 90-90-45 triangle are in the ratio 1:1:√2, which is a key characteristic of these triangles.
- The triangle's area can be calculated using the formula A = (1/2)ab, where a and b are the lengths of the two equal sides.
- The 90-90-45 triangle has numerous practical applications in fields such as architecture, engineering, and design.
Geometric Properties
The 90-90-45 triangle has several distinct geometric properties that set it apart from other types of triangles. One of its most notable features is the relationship between its side lengths. In a 90-90-45 triangle, the two legs (the sides that meet at a right angle) are equal in length, while the hypotenuse (the side opposite the right angle) is √2 times the length of either leg. This means that if the length of one leg is x, the length of the other leg is also x, and the length of the hypotenuse is x√2.
Side Length Ratios
The side length ratios of a 90-90-45 triangle are a key aspect of its geometric properties. The ratio of the lengths of the two legs to the hypotenuse is 1:1:√2, which is a fundamental property of these triangles. This ratio can be expressed mathematically as:
a : b : c = 1 : 1 : √2
where a and b are the lengths of the two legs, and c is the length of the hypotenuse.
| Side Length | Ratio |
|---|---|
| Leg 1 | 1 |
| Leg 2 | 1 |
| Hypotenuse | √2 |
Trigonometric Properties
The 90-90-45 triangle also has several important trigonometric properties. One of the most significant is the relationship between the sine, cosine, and tangent of the 45-degree angle. In a 90-90-45 triangle, the sine and cosine of the 45-degree angle are equal, and the tangent is equal to 1. This can be expressed mathematically as:
sin(45°) = cos(45°) = 1/√2
tan(45°) = 1
Trigonometric Identities
The trigonometric identities that apply to 90-90-45 triangles are essential for solving problems involving these triangles. Some of the most important identities include:
sin(45°) = cos(45°)
tan(45°) = 1
csc(45°) = sec(45°) = √2
These identities can be used to simplify trigonometric expressions and solve problems involving 90-90-45 triangles.
What is the ratio of the side lengths of a 90-90-45 triangle?
+The ratio of the side lengths of a 90-90-45 triangle is 1:1:√2, where the two legs are equal in length and the hypotenuse is √2 times the length of either leg.
What are the trigonometric properties of a 90-90-45 triangle?
+The trigonometric properties of a 90-90-45 triangle include the relationships between the sine, cosine, and tangent of the 45-degree angle, which are sin(45°) = cos(45°) = 1/√2 and tan(45°) = 1.
What are some practical applications of 90-90-45 triangles?
+90-90-45 triangles have numerous practical applications in fields such as architecture, engineering, and design, where they are used to create structures and designs that are both functional and aesthetically pleasing.
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