Problem 5 - College Algebra Essentials

Question

find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (0,0) \text { and }(-3,4) $$

Step-by-Step Solution

Answer

The distance between the points (0,0) and (-3,4) is 5 units.

Step 1: Identify the coordinates

The given points on the plane are (0,0) and (-3,4). Here, (0,0) can be written as \((x_1, y_1)\) and (-3,4) as \((x_2, y_2)\).

Step 2: Substitute the coordinates into the distance formula

Substitute these coordinates into the distance formula, \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). This results in \(d = \sqrt{(-3 - 0)^2 + (4 - 0)^2}\)

Step 3: Simplify the expression

Next, simplify the expression inside the radical: \(d = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16}\)

Step 4: Evaluation

Evaluate the square root operation to find the distance: \(d = \sqrt{25}\)

Step 5: Round answer to two decimal places

Eventually, take the square root of 25 to get the final answer. As there is no decimal in the square root of 25, no rounding is needed.

Key Concepts

Coordinate Geometry

Coordinate geometry is an important branch of mathematics that combines algebra and geometry. This technique allows you to determine geometric properties and relationships of points, lines, and shapes by using an algebraic approach. It’s particularly helpful when you want to pinpoint the location or calculate the distance between different points on the plane.
Coordinate geometry utilizes a set of numeric coordinates to effectively describe the positions. These coordinates typically exist in pairs, represented as

• (x, y) for a 2D plane
• (x, y, z) for 3D spaces

Each pair precisely marks a unique position on the Cartesian plane, where the x-coordinate specifies its horizontal location and the y-coordinate tells its vertical position.

Distance Calculation

In coordinate geometry, one of the most fundamental operations is calculating the distance between two specific points. To find this distance, we use a specific formula known as the distance formula.
The distance formula is derived from the Pythagorean theorem. It gives you a straightforward way to calculate the linear distance by considering the horizontal and vertical differences between two points. Given two points

• (x₁, y₁)
• (x₂, y₂)

, the distance

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

is obtained by substituting the coordinates into the formula, calculating the differences, squaring them, adding them, and finally taking the square root of the sum. This step-by-step process is easy once you understand it, and is foundational in many areas of geometry and physics.

Cartesian Plane

The Cartesian plane, named after the mathematician René Descartes, is a two-dimensional coordinate system. It is a flat surface that allows anyone to plot points by providing two numbers called coordinates.
It is constructed with:

• a horizontal axis (x-axis)
• a vertical axis (y-axis)

These axes intersect at a point known as the origin, which has coordinates (0,0). Both axes have positive and negative directions, divided by the origin, thus creating four distinct regions called quadrants.
The Cartesian plane is crucial since it gives a visual method to interpret how distance and position relate to one another. This makes it a powerful tool to approach problems in both mathematics and science effectively.