Pythagorean Theorem

Definition

The Pythagorean Theorem is a fundamental relationship in geometry that describes the connection between the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

5 Must Know Facts For Your Next Test

1. The Pythagorean Theorem is often represented by the equation: $a^2 + b^2 = c^2$, where $a$ and $b$ are the lengths of the two shorter sides, and $c$ is the length of the hypotenuse.
2. The Pythagorean Theorem is used to calculate the length of the unknown side in a right triangle, given the lengths of the other two sides.
3. The square roots of irrational numbers, such as $\sqrt{2}$, are often used in the context of the Pythagorean Theorem.
4. The Pythagorean Theorem is a fundamental concept in trigonometry and is used in many applications, including construction, navigation, and physics.
5. The Pythagorean Theorem can be used to find the distance between two points on a coordinate plane, as well as to calculate the area and perimeter of various geometric shapes.

Review Questions

• Explain how the Pythagorean Theorem is used to simplify and use square roots, as described in Section 5.7.
  • The Pythagorean Theorem is closely related to the concept of simplifying and using square roots, as described in Section 5.7. When solving for the length of an unknown side in a right triangle, the Pythagorean Theorem often requires the use of square roots to find the solution. For example, if the lengths of the two shorter sides of a right triangle are known, the Pythagorean Theorem can be used to calculate the length of the hypotenuse, which may involve taking the square root of the sum of the squares of the other two sides.
• Describe how the Pythagorean Theorem is connected to the concept of rational and irrational numbers, as discussed in Section 7.1.
  • The Pythagorean Theorem is closely linked to the concept of rational and irrational numbers, as discussed in Section 7.1. The square roots that often arise when applying the Pythagorean Theorem, such as $\sqrt{2}$, are examples of irrational numbers, which cannot be expressed as a simple fraction. The recognition that the square root of 2 is an irrational number was a significant mathematical discovery, as it challenged the prevailing belief that all numbers could be represented as ratios of integers. This connection between the Pythagorean Theorem and irrational numbers highlights the importance of understanding the properties of different number systems in the context of geometric relationships.
• Analyze how the Pythagorean Theorem is used to solve problems involving angles, triangles, and other geometric properties, as described in Sections 9.3 and 9.4.
  • The Pythagorean Theorem is a fundamental tool for solving a wide range of problems involving angles, triangles, and other geometric properties, as discussed in Sections 9.3 and 9.4. By applying the Pythagorean Theorem, one can determine the lengths of unknown sides in right triangles, which in turn can be used to calculate angles, perimeters, and areas of various shapes. Additionally, the Pythagorean Theorem is essential for understanding the properties of rectangles, triangles, and trapezoids, as it allows for the calculation of diagonal lengths and the identification of right angles within these geometric figures. The versatility of the Pythagorean Theorem makes it a powerful tool for solving a wide range of problems in geometry and related fields.

Related terms

Right Triangle:

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

Hypotenuse:

The longest side of a right triangle, which is opposite the right angle.

Irrational Number:

A number that cannot be expressed as a simple fraction, such as the square root of 2.