Question

State the Distance Formula for points represented by \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\)

Short answer

Distance Formula: \\(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\)

Step-by-step solution

Understanding the Problem

The task is to state the Distance Formula, which is used to calculate the distance between two points in a coordinate plane, represented by their coordinates.

Identify the Formula Components

To derive the formula, recognize that the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) forms the hypotenuse of a right triangle, with the differences in x and y as the other two sides.

Write the Displacement Expressions

These differences are given by \((x_2 - x_1)\) and \((y_2 - y_1)\), representing the horizontal and vertical distances, respectively, between the two points.

Apply the Pythagorean Theorem

According to the Pythagorean Theorem, the square of the hypotenuse (distance \(d\)) is equal to the sum of the squares of the other two sides: \(d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2\).

Solve for the Distance

Taking the square root on both sides to solve for \(d\) gives the Distance Formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Key concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where points are identified by pairs of numbers, known as coordinates. These coordinates are represented by \(x, y\), where \(x\) indicates the horizontal position and \(y\) indicates the vertical position.
The plane itself is divided by two perpendicular lines called axes. The horizontal axis is known as the x-axis, while the vertical axis is referred to as the y-axis. The point where these axes intersect is called the origin, identified by (0, 0). This grid-like network makes it easy to locate any point you may want to find.

• Locating Points: To locate a point like (3, 4), you move three units along the x-axis and four units up the y-axis.
• Positive and Negative Values: Moving to the right or up from the origin involves positive values, while moving left or down involves negative values.

Understanding the coordinate plane is crucial for working with geometric concepts like the Distance Formula, as it helps you visualize the distances between points.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that relates the sides of a right triangle. In simple terms, it states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, it's expressed as \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse, and \(a\) and \(b\) are the triangle's other sides.
This theorem is instrumental in deriving the Distance Formula on a coordinate plane. Let's break this down:

• Imagine you have a right triangle formed between two points on a coordinate plane.
• The differences \(x_2 - x_1\) and \(y_2 - y_1\) represent the two shorter sides of the triangle.
• Applying the Pythagorean Theorem helps determine the length of the hypotenuse, which is the distance between the two points.

By understanding the Pythagorean Theorem, you're equipped to tackle many geometry problems, including calculating distances on a graph.

Right Triangle

A right triangle is a type of triangle that has a 90-degree angle, often depicted as a corner or L-shape. It's an essential shape in the study of geometry because of its direct relationship with the Pythagorean Theorem.
In a right triangle:

• The side opposite the right angle is called the hypotenuse.
• The other two sides are referred to as the legs of the triangle.

Understanding the concept of a right triangle is particularly helpful when working with coordinate geometry and the Distance Formula. When two points on a coordinate plane form a right triangle with the axes, the hypotenuse of this triangle represents the distance between the points.

• Imagine plotting two points and drawing a vertical and horizontal line from each point until they meet at a right angle. This forms our right triangle.
• These lines represent the difference in the x-coordinates and y-coordinates, or the 'legs' of the triangle.

Recognizing this relationship is key when applying the Distance Formula, allowing for an easier interpretation of distances in a coordinate system.