Pythagorean Theorem

Definition

The Pythagorean Theorem is a fundamental relationship in geometry that describes the connection between the lengths of the sides of a right triangle. It states that 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.

5 Must Know Facts For Your Next Test

1. The Pythagorean Theorem can be expressed mathematically as $a^2 + b^2 = c^2$, 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 fundamental to vector addition and subtraction, as it allows for the determination of the magnitude of a vector from its components.
3. In vector addition and subtraction, the Pythagorean Theorem can be used to find the length of the resultant vector from the lengths of the individual vectors.
4. The Pythagorean Theorem is also used to determine the distance between two points in a coordinate system, as the distance can be calculated using the lengths of the sides of the right triangle formed by the two points.
5. The Pythagorean Theorem has numerous applications in physics, engineering, and other scientific fields, as it provides a way to calculate the lengths of unknown sides in right triangles.

Review Questions

• Explain how the Pythagorean Theorem is used in the context of vector addition and subtraction.
  • The Pythagorean Theorem is fundamental to vector addition and subtraction because it allows for the determination of the magnitude of a resultant vector from the lengths of its component vectors. When two vectors are added or subtracted graphically, the resulting vector can be viewed as the hypotenuse of a right triangle, where the component vectors form the other two sides. The Pythagorean Theorem can then be applied to calculate the length of the resultant vector using the lengths of the component vectors.
• Describe how the Pythagorean Theorem can be used to find the distance between two points in a coordinate system.
  • The Pythagorean Theorem can be used to calculate the distance between two points in a coordinate system. The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ can be found by treating the line segment connecting the two points as the hypotenuse of a right triangle, where the other two sides are the horizontal and vertical distances between the points. The Pythagorean Theorem can then be applied to find the length of the hypotenuse, which represents the distance between the two points, using the formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Analyze how the Pythagorean Theorem is used in the analytical methods of vector addition and subtraction.
  • In the analytical methods of vector addition and subtraction, the Pythagorean Theorem is used to determine the magnitude of the resultant vector from the magnitudes and directions of the individual vectors. By representing the vectors in a coordinate system, the components of the vectors can be used to form a right triangle, where the resultant vector is the hypotenuse. The Pythagorean Theorem can then be applied to calculate the length of the resultant vector using the formula $|\vec{R}| = \sqrt{(\vec{A}_x + \vec{B}_x)^2 + (\vec{A}_y + \vec{B}_y)^2}$, where $\vec{A}$ and $\vec{B}$ are the individual vectors, and $\vec{R}$ is the resultant vector.

Related terms

Right Triangle: A triangle with one 90-degree angle, where the side opposite the right angle is called the hypotenuse.

Hypotenuse: The side of a right triangle that is opposite the right angle and is the longest side of the triangle.

Trigonometry: The branch of mathematics that deals with the relationships between the sides and angles of triangles.