Problem 55 - Introductory Algebra for College Students

Question

Find the distance between each pair of points. Express answers in simplified radical form and, if necessary, round to two decimal places. $$(-7,-5) \text { and }(-2,-1)$$

Step-by-Step Solution

Answer

The distance between the points (-7, -5) and (-2, -1) is \[\sqrt{41}\]

Step 1: Identify the Points

The two points provided are (-7, -5) and (-2, -1). Here, (-7, -5) is point 1 and (-2, -1) is point 2.

Step 2: Apply the Distance Formula

Using the distance formula, the distance between the two points is calculated as \[\sqrt{((-2)-(-7))^2+((-1)-(-5))^2}\] This simplifies to \[\sqrt{(5)^2+(4)^2}\]

Step 3: Simplify the Equation

Simplify the equation to get \[\sqrt{(25)+(16)}\] Further simplification leads to \[\sqrt{41}\]

Key Concepts

Distance Formula

Understanding how to determine the distance between two points in a plane is an essential aspect of coordinate geometry. Our main tool for this task is the distance formula, which is grounded in the Pythagorean theorem. The distance formula is expressed as \(d=\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\), where \(x_1,y_1\) and \(x_2,y_2\) are the coordinates of the two points.

To use it effectively, simply take the differences of the x-coordinates and y-coordinates separately, square each result, add them together, and then take the square root of the sum. This formula provides the linear distance 'as the crow flies' between any two points in a coordinate plane.

Radical Simplification

After calculating the distance using the distance formula, you will often end up with a radical expression. Radical simplification is the process of making these expressions as concise as possible. A radical, often called the square root, can be simplified by identifying and extracting perfect squares from the expression.

In the provided exercise, the radical \(\sqrt{41}\) cannot be simplified further because 41 is not a perfect square. A perfect square is a number that can be expressed as the product of an integer with itself, such as 16, which is 4 times 4. If the expression under the radical had been 40 instead of 41, for example, we could have simplified \(\sqrt{40}\) to 2 times \(\sqrt{10}\), as 40 is 4 (a perfect square) times 10. The goal of radical simplification is to make the expression more readable and easier to work with, especially when incorporating it into further calculations or graphical interpretations.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that allows us to use algebra to solve geometric problems. By plotting points, lines, and shapes on a coordinate plane, we can apply algebraic formulas to find distances, midpoints, slopes, and more.

The coordinate plane is divided into four quadrants by a horizontal x-axis and a vertical y-axis. Each point on this plane has a pair of coordinates: the first number (x-coordinate) measures the horizontal distance from the origin, and the second number (y-coordinate) measures the vertical distance.

Importance of Quadrants

Determining which quadrant a point is located in can help us understand the relationship between points and simplify calculations, particularly when dealing with distance and midpoint formulas. In the context of our exercise, knowing the quadrant is not crucial as the distance formula is quadrant-agnostic, but it's still a valuable piece of geometric intuition.