Problem 18 - College Algebra with Corequisite Support

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. (2,-5) and (7,4)

Step-by-Step Solution

Answer

The distance is \(\sqrt{106}\) in simplest radical form.

Step 1: Write the Distance Formula

To find the distance between two points \(x_1, y_1\) and \(x_2, y_2\), use the distance formula: \\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] \Here, \(x_1 = 2\), \(y_1 = -5\), \(x_2 = 7\), and \(y_2 = 4\).

Step 2: Substitute the Coordinates into the Formula

Substitute the given coordinates \(x_1 = 2\), \(y_1 = -5\), \(x_2 = 7\), and \(y_2 = 4\) into the distance formula: \\[D = \sqrt{(7 - 2)^2 + (4 + 5)^2}\]

Step 3: Calculate the Differences

Calculate the differences: \(7 - 2 = 5\) and \(4 - (-5) = 9\).

Step 4: Square the Differences

Square each of the differences: \(5^2 = 25\) and \(9^2 = 81\).

Step 5: Add the Squares

Add the squares from the previous step: \(25 + 81 = 106\).

Step 6: Find the Square Root

Calculate the square root of \(106\) to determine the distance: \(\sqrt{106}\). 106 is simplified as it does not contain any perfect square factors other than 1, making \(\sqrt{106}\) the simplest radical form.

Key Concepts

Simplest Radical Form

When calculating the distance between two points, sometimes your answer will be an irrational number that doesn't have a whole number output. In these situations, it's best to express your answer in its simplest radical form. This means simplifying the square root so that it has the smallest possible integer outside the radical sign.

For example, when you calculate the distance between the points (2, -5) and (7, 4), you end up with \[ \sqrt{106} \]. Since 106 does not have any perfect square factors other than 1, it cannot be simplified further.

Therefore, \[ \sqrt{106} \] is already in its simplest radical form. This method helps keep your calculations precise, especially when dealing with unusual or lengthy decimals.

Calculate Differences

The first step in using the Distance Formula is to calculate the differences in the x-coordinates and the y-coordinates of the given points.

• The difference in the x-coordinates is \( x_2 - x_1 \). In our problem, this is \( 7 - 2 \), which equals 5.
• The difference in the y-coordinates is \( y_2 - y_1 \). For our points, this is \( 4 - (-5) \), resulting in a difference of 9.

By calculating these differences, you set the groundwork for finding the distance between the points. It's a simple but crucial step to get accurate results.

Square the Differences

After calculating the differences between the coordinates, the next step is to square those differences. Squaring a number means multiplying the number by itself.

Squaring serves to eliminate any negative signs that might occur in differences:

• The square of the x-difference, \( 5^2 \), equals 25.
• The square of the y-difference, \( 9^2 \), equals 81.

This step will help us later in summing these squared numbers, crucial for the final calculation of the distance using the Distance Formula. Squaring ensures all parts of the equation are positive, maintaining the correct calculation pattern in geometry.