Problem 6 - Algebra 2

Question

Find the distance between each pair of points with the given coordinates. $$ (7,8),(-4,9) $$

Step-by-Step Solution

Answer

Short answer: The distance is approximately 11.05.

Step 1: Apply the distance formula

\(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} = \sqrt{(7-(-4))^2 + (8-9)^2}\)

Step 2: Compute

\(d = \sqrt{11^2 + (-1)^2} = \sqrt{121 + 1} = \sqrt{122} \approx 11.05\)

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a field of mathematics that uses a coordinate system to define and evaluate geometrical shapes and their properties. By utilizing a coordinate plane with horizontal and vertical axes (commonly x and y), you can uniquely identify each point in space using ordered pairs. In the case of two-dimensional space, each ordered pair consists of two numbers, representing the x-coordinate and y-coordinate of a point.

For example, the point (7, 8) resides at 7 units along the x-axis and 8 units up the y-axis. Similarly, the point (-4, 9) is located 4 units left from the origin and 9 units upwards. Coordinate geometry provides a meaningful way to represent space, calculate distances, and analyze the relative positions of points, which is highly beneficial in mathematics and various fields such as physics and engineering.

Distance Calculation

Calculating the distance between two points in a two-dimensional space is a fundamental application of coordinate geometry. The main tool used here is the distance formula, which derives from the Pythagorean theorem. Imagine drawing a right triangle with the two points as its vertices. The horizontal and vertical distances between the points form the legs of the triangle, while the distance we seek is the hypotenuse.

The formula for calculating the distance (\(d\)) is:

• Subtract the x-coordinates: (\(x_2 - x_1\))
• Subtract the y-coordinates: (\(y_2 - y_1\))
• Square the results of both differences
• Add these squares
• Take the square root of the sum

For example, for the points (7, 8) and (-4, 9), the calculations are:

• \(x_2 - x_1 = -4 - 7 = -11\)
• \(y_2 - y_1 = 9 - 8 = 1\)
• Distance \(d = \sqrt{(-11)^2 + (1)^2} = \sqrt{121 + 1} = \sqrt{122} \approx 11.05\)

This formula simplifies distance computation in a straightforward manner, providing clarity and precision in quantitative analysis.

Two-Dimensional Space

A two-dimensional space is commonly represented on a flat plane with two axes—usually labeled x for horizontal and y for vertical directions. Each point in this plane can be described using two numbers: its horizontal position on the x-axis, and its vertical position on the y-axis. This simple system allows for complex ideas and geometrical relationships to be expressed and explored.

Understanding two-dimensional space is crucial for solving practical problems involving distance and position. For the task of finding the distance between two points, the 2D concept allows us to apply the distance formula effectively.

In essence, *two-dimensional space* provides the foundation to geometrically interpret and visually represent elements such as lines, curves, and distances, essential skills in mathematics, cartography, computer graphics, and even in navigating our day-to-day environments.