Problem 9 - College Algebra

Question

Find the distance between each pair of points. If necessary, round answers to two decimals places. $$(0,-3)\( and \)(4,1)$$

Step-by-Step Solution

Answer

The distance between the points (0, -3) and (4, 1) is approximately 5.66 units.

Step 1: Identify the Coordinates

Identify and label the coordinates of the two points. Here, (x1, y1) = (0, -3) and (x2, y2) = (4, 1).

Step 2: Apply Distance Formula

Apply the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) by replacing x1, y1, x2, and y2 with the coordinates identified in step 1. The result is \(d = \sqrt{(4 - 0)^2 + (1 - (-3))^2}\).

Step 3: Simplify The Equation

Simplify the equation from step 2. \(d = \sqrt{(4)^2 + (4)^2} = \sqrt{16 + 16}\).

Step 4: Final Calculation

Finish the calculation to receive \(d = \sqrt{32} = 5.66\) after rounding up to two decimals.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric figures through a coordinate system. By using coordinates, such as (x, y), we can precisely locate a point on a plane. This system allows us to represent geometric figures, find distances, and much more.
In our given exercise, we have two points:

• (0, -3)
• (4, 1)

The first number in each pair represents the 'x-coordinate', while the second represents the 'y-coordinate'. By identifying these coordinates, we can plot the points onto a cartesian plane, which gives us a visual representation of their positions.
This visualization aids us in applying mathematical formulas to solve problems, such as calculating the distance between two points.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in mathematics used to determine the length of the sides in a right-angled triangle. It states that in such a triangle, the square of the hypotenuse, or the longest side, is equal to the sum of the squares of the other two sides. This theorem is often expressed in the equation: \[ a^2 + b^2 = c^2 \]where \(c\) represents the hypotenuse.
In the context of coordinate geometry, this principle is used to derive the distance formula. When you want to find the distance between two points on a plane, you can imagine drawing a right triangle with these points.
The horizontal leg represents the difference in x-coordinates, \((x_2 - x_1)\), and the vertical leg is the difference in y-coordinates, \((y_2 - y_1)\). By applying the Pythagorean theorem, the distance between the two points is represented by the hypotenuse: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Thus, this formula is simply an application of the Pythagorean theorem to the coordinate plane.

Equation Simplification

Equation simplification is the process of making an equation easier to solve or understand. In mathematical problems, particularly those involving calculations, simplifying equations is crucial. It involves reducing expressions and combining terms. Simplification ensures calculations are straightforward and helps in getting to the solution without unnecessary complexity.
In our solution, after applying the distance formula, the equation included expressions such as:

• \((x_2 - x_1)^2\)
• \((y_2 - y_1)^2\)

Step by step, we simplify these, first by calculating the squared values:

• For the x-coordinates: \((4 - 0)^2 = 16\)
• For the y-coordinates: \((1 - (-3))^2 = 16\)

Adding these squares gives us 32, and finally, finding the square root provides us our distance of 5.66 (rounded to two decimal places). Simplifying helps us not only find answers but understand the path we take to reach them.