Problem 4 - College Algebra

Question

Find the distance between each pair of points. If necessary, round answers to two decimals places. $$(2,-3)\( and \)(-1,5)$$

Step-by-Step Solution

Answer

The distance between the points (2,-3) and (-1,5) is approximately 8.54 units.

Step 1: Identify the coordinates

Identify the coordinates of the given points. Here, the coordinates of the first point are (2,-3) and that of the second point are (-1,5).

Step 2: Apply the Distance Formula

Substitute these values into the distance formula. We get \[\sqrt{((-1)-2)^2 + (5-(-3))^2} = \sqrt{(-3)^2 + (5+3)^2}\].

Step 3: Simplify

Simplify the expression inside the square root to get \[\sqrt{9 + 64}\].

Step 4: Calculate

Perform the calculation to get the square root of 73 which is approximately 8.54 when rounded to two decimal places.

Key Concepts

Coordinate Geometry

Coordinate geometry, sometimes referred to as analytic geometry, is a branch of mathematics that describes geometric figures using an ordered system of coordinates. In this context, when we talk about points on a plane, we use a pair of numerical values to represent a specific location. Each point is defined by two coordinates:

• The first value is the x-coordinate, which shows the position along the horizontal axis.
• The second value is the y-coordinate, indicating the position along the vertical axis.

For the points in our problem, we have (2,-3) and (-1,5). These numbers tell us exactly where the points are on a grid or plane. To really understand how far apart these points are, we need to consider their coordinates in a structured way, which leads us to use the Distance Formula.

Distance Calculation

To calculate the distance between two points in a plane, we utilize the Distance Formula, a fundamental tool in coordinate geometry. This formula is derived from the Pythagorean theorem, which is widely used to find the lengths of sides in a right triangle. The formula is:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]where:

• \(x_1, y_1\) are the coordinates of the first point.
• \(x_2, y_2\) are the coordinates of the second point.

In our specific problem, substituting the values from points (2,-3) and (-1,5) into the formula, we get:\[\sqrt{((-1) - 2)^2 + (5 - (-3))^2} = \sqrt{(-3)^2 + (8)^2}\]Calculating these differences gives us the squares \((-3)^2\) and \(8^2\), creating the foundation for our final distance.

Square Root Simplification

The last step in solving this distance problem involves simplifying the expression under the square root. After substituting the coordinates and calculating the individual differences, we reach the equation:\[\sqrt{9 + 64}\]This simplification is crucial to finding the distance. Here, we get:

• \(9\) from \((-3)^2\)
• \(64\) from \(8^2\)

Adding these values together gives us a total of \(73\). The final task is to compute the square root of this sum. The square root of \(73\) is approximately \(8.54\), which is rounded to two decimal places as required by the problem. Simplifying the square root not only helps in accurate calculations but involves understanding the arithmetic of squares and roots, which is a fundamental process in many areas of mathematics.