Distance Formula

In this chapter, we will explore the Distance Formula, which is a fundamental concept in coordinate geometry. This formula allows us to calculate the distance between two points in a Cartesian plane. Understanding this formula is crucial for solving various problems related to geometry, physics, and real-life applications.

Key Concepts

1. Cartesian Plane

The Cartesian Plane is a two-dimensional plane formed by the intersection of two perpendicular lines called the x-axis and the y-axis. The point where these axes intersect is called the origin, denoted as (0, 0). Points on this plane are represented as ordered pairs (x, y).

2. Distance Between Two Points

The distance between two points in the Cartesian Plane can be calculated using the Distance Formula. If we have two points:

• Point A with coordinates (x₁, y₁)
• Point B with coordinates (x₂, y₂)

The Distance Formula is given by:

[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

Where:

• d represents the distance between the two points.
• (x₁, y₁) and (x₂, y₂) are the coordinates of points A and B, respectively.

3. Derivation of the Distance Formula

To derive the Distance Formula, we can use the Pythagorean Theorem. Consider a right triangle formed by the two points A and B, where the base and height are the differences in the x-coordinates and y-coordinates, respectively.

1. The length of the base is |x₂ - x₁|.
2. The length of the height is |y₂ - y₁|.

According to the Pythagorean Theorem: [ c^2 = a^2 + b^2 ] Where:

• c is the hypotenuse (the distance d),
• a is the base, and
• b is the height.

Substituting the lengths of the base and height, we have: [ d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2 ] Taking the square root of both sides gives us the Distance Formula: [ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

4. Examples of Using the Distance Formula

Example 1: Basic Calculation

Find the distance between the points (3, 4) and (7, 1).

Using the Distance Formula:

• Here, (x₁, y₁) = (3, 4) and (x₂, y₂) = (7, 1).
• Calculate: [ d = \sqrt{(7 - 3)^2 + (1 - 4)^2} ] [ d = \sqrt{(4)^2 + (-3)^2} ] [ d = \sqrt{16 + 9} ] [ d = \sqrt{25} ] [ d = 5 ]

Thus, the distance between the points (3, 4) and (7, 1) is 5 units.

Example 2: Negative Coordinates

Find the distance between the points (-2, -3) and (4, 1).

Using the Distance Formula:

• Here, (x₁, y₁) = (-2, -3) and (x₂, y₂) = (4, 1).
• Calculate: [ d = \sqrt{(4 - (-2))^2 + (1 - (-3))^2} ] [ d = \sqrt{(4 + 2)^2 + (1 + 3)^2} ] [ d = \sqrt{(6)^2 + (4)^2} ] [ d = \sqrt{36 + 16} ] [ d = \sqrt{52} ] [ d = 2\sqrt{13} ]\

Thus, the distance between the points (-2, -3) and (4, 1) is 2√13 units.

Summary

In this chapter, we learned about the Distance Formula and how to calculate the distance between two points in a Cartesian plane. We derived the formula using the Pythagorean Theorem and worked through examples to solidify our understanding. The Distance Formula is a powerful tool that can be applied in various fields, including mathematics, physics, and engineering. Mastering this concept will greatly enhance your problem-solving skills in geometry and beyond.