Problem 2 - College Algebra Essentials

Question

find the distance between each pair of points. If necessary, round answers to two decimals places. $$(5,1) \text { and }(8,5)$$

Step-by-Step Solution

Answer

The distance between the points (5,1)and (8,5) is 5

Step 1: Identifying the coordinates

Identify the coordinates of each point. Let \((x_1, y_1)\) be (5,1) and \((x_2, y_2)\) be (8,5).

Step 2: Plugging the coordinates into the formula

Insert the identified values into the distance formula. Then the distance \(d\) becomes \(d = \sqrt{((8-5)^2 + (5-1)^2)}\)

Step 3: Calculating the distance

Now need to simplify the expression above. The distance \(d\) will then be \(d=\sqrt{(3)^2 + (4)^2} = \sqrt{9+16} = \sqrt{25} = 5 \)

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that deals with the study of geometric figures using a coordinate system. It's like connecting the dots on a map, where each point is defined by a pair of numbers: the x-coordinate and the y-coordinate. These pairs, called coordinates, are like GPS coordinates on a plane. They tell us where a point is located.

In our exercise, we have two points: (5,1) and (8,5). The first number in each pair is the x-coordinate, which tells us how far along the horizontal axis the point is. The second number is the y-coordinate, showing us the position along the vertical axis. By plotting these points on a graph, you can visualize the line between them.

Understanding coordinates is essential as they give us a way to measure distances and create shapes, helping us solve geometric problems like the one in this exercise.

Euclidean Distance

The concept of Euclidean distance comes from the idea of finding the shortest path between two points on a flat surface. This distance is calculated using the distance formula, which is derived from the principles of coordinate geometry and the Pythagorean theorem. The Euclidean distance metric measures the 'straight-line' distance, as you would measure between two points using a ruler.

For any two points, such as \(x_1, y_1\) and \(x_2, y_2\), the Euclidean distance is given by the formula:

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

In our example, the points (5,1) and (8,5) use this formula to determine the distance between them. You replace \(x_1,y_1\) with (5,1) and \(x_2,y_2\) with (8,5), simplifying the equation to find that the distance is 5 units.

Understanding Euclidean distance is crucial because it's widely used in various fields such as physics, computer science, and geography whenever you need to calculate the shortest distance between two points.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that helps to find the length of sides in a right-angled triangle. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

This theorem can be written as: \(a^2 + b^2 = c^2\), where \(c\) represents the length of the hypotenuse, while \(a\) and \(b\) are the lengths of the other two sides.

In our case, when determining the distance between two points, the differences in x and y coordinates form the two shorter sides of a right-angled triangle, with the line connecting the points acting as the hypotenuse. According to the exercise, when applying the Pythagorean theorem to the coordinates (5,1) and (8,5), we compute:

• \(3^2 + 4^2 = 5^2\)

This confirms that the calculated distance of 5 units aligns perfectly with the theorem. Thus, this method allows you to compute the "straight line" between two points on a flat plane, reinforcing the calculation done using the Euclidean distance formula.