Problem 11 - College Algebra

Question

Find the distance between each pair of points. If necessary, round answers to two decimals places. $$(3.5,8.2)\( and \)(-0.5,6.2)$$

Step-by-Step Solution

Answer

The distance between the points (3.5, 8.2) and (-0.5, 6.2) is approximately 4.47 units.

Step 1: Identify the coordinates

Identify and label the coordinates. Let's label them as follows: \((x_1, y_1) = (3.5, 8.2)\) and \((x_2, y_2) = (-0.5, 6.2)\)

Step 2: Plug the values into the formula

Now, plug these values into the distance formula: \(d = \sqrt{(-0.5-3.5)^2 + (6.2-8.2)^2}\)

Step 3: Calculate the square of each number

Calculate the squares inside brackets: \(d = \sqrt{(-4)^2 + (-2)^2}= \sqrt{16+4}\)

Step 4: Sum up and take the square root

Now sum up these squares and then take the square root: \(d = \sqrt{20} = 4.47\) (rounded to two decimal places).

Key Concepts

Coordinate Geometry Basics

Coordinate geometry, also known as analytic geometry, deals with defining and representing geometric shapes in a numerical way. This system is based on a coordinate plane, which is a two-dimensional space defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Any point in this space can be described using a pair of coordinates, like \(x,y\), where 'x' represents the position along the x-axis, and 'y' represents the position along the y-axis.
This allows us to precisely locate points and measure distances between them, leading to a deeper understanding of spatial relationships. In the problem given, the points (3.5, 8.2) and (-0.5, 6.2) are defined in this coordinate system. The goal is to find the distance between these two points using coordinate geometry principles.

Understanding Distance Calculation

Finding the distance between two points on a plane involves using the distance formula, an essential tool in coordinate geometry. The distance formula is derived from the Pythagorean theorem and is written as:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
This formula calculates the straight-line distance (also known as Euclidean distance) between two points by summing the squares of the differences in their x and y coordinates, then taking the square root of that sum.

• The x-coordinates from each point are subtracted, squared, and added to the square of the difference in the y-coordinates.
• Finally, the square root of this total gives the final distance.

This method ensures accuracy and consistency in measuring distances, which is especially useful in fields like physics and engineering.

Rounding Decimals

After calculating a numerical answer, it's often necessary to round the result to make it more readable or to meet specific precision requirements. In the context of the given problem, the distance is calculated to be \[ \sqrt{20} \]. When you compute this value, it equals approximately 4.4721359. For practical purposes, we might need to round this to a number with fewer decimal places, such as two decimal places.

• To round to two decimal places, look at the third decimal place.
• If it's 5 or greater, you round the second decimal place up. If it's less than 5, you keep the second decimal place as is.

For \sqrt{20}\, the third decimal is 2, which means the correct rounded result to two decimal places is 4.47. Rounding helps simplify data presentation without significantly losing accuracy.