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Chapter 9: Problem 83

State the Pythagorean theorem in words.

Short Answer

Expert verified
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

Step by step solution

01

Introduction to the Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, particularly used in the study of right triangles. It expresses the relationship between the lengths of the sides of a right triangle.
02

Understanding Right Triangles

A right triangle consists of three sides: two legs and the hypotenuse. The two legs form the right angle, and the hypotenuse is the side opposite the right angle.
03

The Theorem's Statement in Words

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Right Triangles
Right triangles are a special type of triangle that play a key role in the field of geometry. They are defined by having one of their angles as a right angle, exactly 90 degrees. This right angle is what gives the right triangle its name.
  • The two sides that create the right angle are known as the "legs" of the triangle.
  • The side opposite the right angle is called the "hypotenuse."
Understanding the structure of right triangles is essential because it forms the foundation for the Pythagorean theorem, which helps solve various geometric problems involving these triangles.
The Role of Geometry in Right Triangles
Geometry, as an area of mathematics, focuses on the properties and relations of points, lines, surfaces, and solids. A critical part of this field is the study of triangles, especially right triangles.
  • Right triangles are used in numerous practical applications, from architecture to engineering.
  • They help us understand spatial relationships and solve complex problems using straightforward calculations.
One of the significant geometric principles involving right triangles is the Pythagorean theorem. This theorem offers a reliable method to calculate the unknown side of a triangle, providing solutions that are precise and practical.
The Hypotenuse and Its Importance
The hypotenuse is the longest side of a right triangle, directly opposite the right angle. Its significance stems from its role in the Pythagorean theorem, which is stated as follows:
For any right triangle, the square of the hypotenuse (\[c\]) is equal to the sum of the squares of the other two sides (\[a\] and \[b\]): \[c^2 = a^2 + b^2\].
By knowing the lengths of the legs, one can easily calculate the hypotenuse using this formula. Similarly, if the hypotenuse and one leg are known, the other leg's length can be determined. This theorem is a powerful tool not only in pure mathematics but also in real-world problem-solving scenarios.

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Most popular questions from this chapter

Use rational exponents to simplify each radical. All variables represent positive real numbers. See Example 10 . $$ \sqrt[8]{q^{2}} $$

Simplify each expression. All variables represent positive real numbers. a. \(\left(64 a^{4}\right)^{1 / 2}\) b. \(\left(64 a^{4}\right)^{-1 / 2}\) c. \(-\left(64 a^{4}\right)^{1 / 2}\) d. \(\frac{1}{\left(64 a^{4}\right)^{1 / 2}}\)

Simplify each expression. All variables represent positive real numbers. $$ \left(m^{4}\right)^{1 / 2} $$

Use a calculator to solve each problem. Round answers to the nearest tenth. The Grand Canyon. The time \(t\) (in seconds) that it takes for an object to fall a distance of \(s\) feet is given by the formula \(t=\frac{\sqrt{s}}{4}\) In some places, the Grand Canyon is one mile ( \(5,280\) feet) deep. How long would it take a stone dropped over the edge of the canyon to hit bottom?

Use a calculator to solve each problem. Round answers to the nearest tenth. Embroidery. The radius \(r\) of a circle is given by the formula \(r=\sqrt{\frac{A}{\pi}},\) where \(A\) is its area. Find the diameter of the embroidery hoop if there are 38.5 in. \(^{2}\) of stretched fabric on which to embroider.
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