Hypotenuse

Definition

The hypotenuse is the longest side of a right triangle, the side opposite the right angle. It is a crucial concept in geometry and trigonometry, particularly when studying the Pythagorean Theorem and simplifying square roots.

5 Must Know Facts For Your Next Test

1. The hypotenuse is always the longest side of a right triangle, and it is the side opposite the 90-degree angle.
2. The Pythagorean Theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
3. Simplifying square roots often involves using the Pythagorean Theorem to find the length of the hypotenuse, given the lengths of the other two sides.
4. The trigonometric functions, such as sine, cosine, and tangent, are defined in terms of the ratios of the sides of a right triangle, including the hypotenuse.
5. The length of the hypotenuse is a crucial factor in many real-world applications, such as engineering, construction, and navigation.

Review Questions

• Explain how the hypotenuse is related to the Pythagorean Theorem and its application in simplifying square roots.
  • The Pythagorean Theorem states 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. This relationship can be used to simplify square roots by finding the length of the hypotenuse given the lengths of the other two sides. For example, if the lengths of the legs of a right triangle are 3 and 4, the length of the hypotenuse can be calculated using the Pythagorean Theorem as $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$. This allows for the simplification of the square root of 25 to the single value of 5.
• Describe how the properties of the hypotenuse are used in trigonometry and the calculation of trigonometric functions.
  • The hypotenuse is a crucial component in the study of trigonometry, as the trigonometric functions (sine, cosine, and tangent) are defined in terms of the ratios of the sides of a right triangle, including the hypotenuse. For example, the cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. Understanding the properties of the hypotenuse, such as its relationship to the other sides of the triangle, is essential for accurately calculating trigonometric functions and solving problems involving right triangles.
• Analyze the importance of the hypotenuse in real-world applications, and explain how its properties are utilized in various fields.
  • The hypotenuse plays a crucial role in many real-world applications, particularly in fields such as engineering, construction, and navigation. Its length is a key factor in determining the stability and strength of structures, the trajectory of projectiles, and the efficiency of transportation systems. For instance, in the design of bridges or buildings, engineers must consider the length of the hypotenuse to ensure the structure's integrity and load-bearing capacity. Similarly, in navigation, the hypotenuse is used to calculate the shortest distance between two points, which is essential for efficient route planning and travel. Understanding the properties of the hypotenuse, such as its relationship to the Pythagorean Theorem and trigonometric functions, allows professionals in these fields to make accurate calculations and informed decisions that have a significant impact on the safety and effectiveness of their work.

Related terms

Right Triangle: A triangle with one 90-degree angle, where the two sides that meet at the right angle are called the legs, and the longest side is the hypotenuse.

Pythagorean Theorem: A fundamental theorem in geometry that states the relationship between the lengths of the three sides of a right triangle, where the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Trigonometry: The study of the relationships between the sides and angles of triangles, which relies heavily on the properties of the hypotenuse and other triangle components.