Problem 8 - College Algebra

Question

Find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (-4,-1) \text { and }(2,-3) $$

Step-by-Step Solution

Answer

The distance between points (-4, -1) and (2, -3) is approximately 6.32.

Step 1: Identify the coordinates

Identify the coordinates of the two points. The first point has coordinates (-4, -1) and the second point has coordinates (2, -3). So, \(x_{1} = -4\), \(y_{1} = -1\), \(x_{2} = 2\), and \(y_{2} = -3\).

Step 2: Substitution into the formula

Substitute the coordinates into the distance formula \(d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}\). This gives us: \(d = \sqrt{(2 - (-4))^{2} + (-3 - (-1))^{2}}\).

Step 3: Simplify the expression

Simplify the expression inside the square root: \(d = \sqrt{(6)^{2} + (-2)^{2}} = \sqrt{36 + 4}\).

Step 4: Calculate the distance

Finally, compute the square root to find the distance: \(d = \sqrt{40} = 6.32\) to two decimal places.

Key Concepts

Cartesian Coordinates

The concept of Cartesian coordinates is vital in understanding distance calculations in mathematics. A Cartesian coordinate system allows us to pinpoint the exact location of any point on a plane using a pair of numerical values. These values are called coordinates and typically denoted as \(x, y\). The first number represents the horizontal position (along the x-axis), while the second represents the vertical position (along the y-axis).
This coordinate system is essential in geometry as it enables the plotting of points and the examination of their relationships with each other. For example, the two points (-4, -1) and (2, -3) lie in this plane with distinct coordinates.
Understanding how to work with Cartesian coordinates is the foundation for solving geometry problems like finding distances. Each point's precise position is determinant in setting up various mathematical operations, making this knowledge a critical piece of the puzzle in performing accurate calculations in coordinate geometry.

Square Root Calculation

Square root calculation is a fundamental operation in mathematics, often used in various formulas, including the distance formula. To find the square root of a number is to determine what value, when multiplied by itself, will yield the original number.
In the context of finding the distance between two points, once we have simplified the expression inside the root, as in \(d = \sqrt{36 + 4}\), the next step is calculating the square root of this sum. This results in \(d = \sqrt{40}\).
For practical purposes, decimals are often rounded to a certain precision, ensuring easy readability and usability. Here, \(\sqrt{40}\) is approximately 6.32 when rounded to two decimal places. Mastering square root calculations is crucial for solving real-world problems that involve measurement and scaling.

Coordinate Geometry

Coordinate Geometry is the branch of mathematics that uses algebraic equations to describe geometric principles involving Cartesian coordinates. It provides tools to answer questions about distances, shapes, angles, and more, by placing geometric figures in the coordinate plane.
The distance formula, for instance, allows us to calculate the precise distance between any two points by substituting their coordinates. This is based on the Pythagorean theorem, where the distance \(d\) between two points \( (x_{1}, y_{1}) \) and \( (x_{2}, y_{2}) \) is derived from: \[ d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}} \].
This formula essentially measures the length of the hypotenuse of a right triangle formed by the points and their vertical and horizontal differences. Mastery of coordinate geometry redefines how we understand the spatial relationships between points. It's not just about drawing; it's about numerical precision in describing and analyzing space.