Problem 20 - College Algebra

Question

For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers. $$(-4,3)\text { and }(10,3)$$

Step-by-Step Solution

Answer

The distance is 14.

Step 1: Identify the Coordinates

The coordinates of the first point are given as (-4, 3), and the coordinates of the second point are (10, 3). Each point has an x-coordinate and a y-coordinate.

Step 2: Use the Distance Formula

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]In this case, substitute \(x_1 = -4\), \(y_1 = 3\), \(x_2 = 10\), and \(y_2 = 3\) into the formula.

Step 3: Simplify the Expression

Substituting the coordinates, we have:\[d = \sqrt{(10 - (-4))^2 + (3 - 3)^2}\]Simplify the terms inside the square root:\[d = \sqrt{(10 + 4)^2 + 0}\]\[d = \sqrt{14^2}\]

Step 4: Solve for the Distance

Calculate the square of 14:\[d = \sqrt{196}\]Finally, take the square root of 196 to find the distance:\[d = 14\]

Key Concepts

Coordinates

To determine the distance between two points in a plane, we start by understanding what coordinates are. Coordinates are a pair of numbers that give the precise location of a point on a graph. They are generally written as (x, y), where *x* is the horizontal position and *y* is the vertical position.
For example, in our problem, we have the points (-4, 3) and (10, 3). Here, the *x*-coordinate tells us how far left or right the point is, and the *y*-coordinate tells us how high or low the point is. Since both points share the same *y*-coordinate (3), they align horizontally, making it a simple scenario to calculate the distance directly along the x-axis.
By understanding coordinates, we can use them to apply formulas like the distance formula to find how far apart two points are.

Simplest radical form

When working with distances, especially in textbook problems, you may encounter solutions asking to provide an answer in "simplest radical form". This concept is useful for expressing roots in a simplified manner, especially when the root value is not a perfect square.
The idea is to present the square root part of a number as simply as possible. Let's say the distance formula gives you something like \(\sqrt{50}\). Instead of leaving it as is, you simplify it. Break down the number inside the square root into prime factors (\(50 = 2 \times 5 \times 5\)), and simplify to \(5\sqrt{2}\). This results in a clearer and neater answer without a calculator's decimal approximation.
However, in the given example, we computed \(\sqrt{196}\), which simplifies neatly to 14, because 196 is a perfect square. It shows that being asked to simplify doesn't always require a radical answer; sometimes it results in a whole number.

Distance between points

Calculating the distance between points is achieved using the distance formula, which is derived from the Pythagorean theorem. This is a crucial concept in geometry and provides a straightforward way to measure straight-line distance between two locations on a grid.
The formula is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Here, - \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points- \( d \) represents the distance between these points. For our example, we had points (-4, 3) and (10, 3), which when applied to the formula result in:\[ d = \sqrt{(10 - (-4))^2 + (3 - 3)^2} \] Upon simplifying this, the formula reduces to \( d = \sqrt{196} \). Since the points only differ in their *x*-coordinates, it's like measuring a straight line on the horizontal axis of a graph. Thus, the distance is 14.
This exercise showcases how the distance formula applies even when movement is purely horizontal or vertical, reminding us that math accurately represents situations we might envision more intuitively.