Problem 13 - College Algebra with Modeling and Visualization

Question

Find the exact distance between the two points. Where appropriate, also give approximate results to the nearest hundredth. $$ (2,-2),(5,2) $$

Step-by-Step Solution

Answer

The exact distance is 5 units.

Step 1: Identify the Coordinates

The two points given are Point 1: \((x_1, y_1) = (2, -2)\)Point 2: \((x_2, y_2) = (5, 2)\)

Step 2: Use the Distance Formula

The formula to find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Step 3: Substitute the Coordinates

Substitute the coordinates of the points into the distance formula:\[ \text{Distance} = \sqrt{(5 - 2)^2 + (2 - (-2))^2} \]

Step 4: Simplify Inside the Square Root

Calculate the differences and squares:\(5 - 2 = 3\) and \(2 - (-2) = 4\)\[ \text{Distance} = \sqrt{3^2 + 4^2} \] \[ \text{Distance} = \sqrt{9 + 16} \] \[ \text{Distance} = \sqrt{25} \]

Step 5: Compute the Final Distance

Calculate the square root to find the exact distance:\(\sqrt{25} = 5\)So, the exact distance is 5.

Key Concepts

Coordinate Geometry

Understanding the concept of coordinate geometry is essential when dealing with point locations on a plane. This method uniquely describes each point using a pair of numerical coordinates. These two numbers give us precise information about a point's position on a flat surface by using an ordered pair, commonly written as \((x, y)\). This pair tells us where the point sits in relation to a fixed grid.

• The first number, known as the x-coordinate, tells us how far left or right the point is.
• The second number, the y-coordinate, tells us how far up or down the point is.

In the exercise provided, two specific points were given on this grid: \((2, -2)\) and \((5, 2)\). These points are located in different quadrants, which demonstrates how coordinate geometry allows us to distinctly identify and work with any point on a plane.

Distance Calculation

Finding the distance between two points in coordinate geometry involves using the distance formula, a straightforward formula derived from the Pythagorean theorem. The formula helps us calculate the true length - or "distance" - that separates two points on the plane. The formula used is:\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here's a simple breakdown of its components:

• \((x_1, y_1)\) are the coordinates of the first point, and \((x_2, y_2)\) are the coordinates of the second point.
• The differences, \((x_2 - x_1)\) and \((y_2 - y_1)\), give us the lengths of the horizontal and vertical sides of the right triangle formed by the points.
• Squaring these differences and adding them accounts for the slant of the line connecting the points.

Applying this formula to our example coordinates, \((5 - 2)^2 + (2 - (-2))^2\), results in:\[ \sqrt{9 + 16} = \sqrt{25} = 5\] This perfect square simplifies the square root operation. Thus, the distance is exact at 5.

Algebraic Expressions

While the distance formula is a key component, understanding algebraic expressions is equally crucial to accurately execute the process. Algebra simplifies the processes and helps translate complex geometric problems into solvable numerical expressions. Here’s how it fits into our example:

• The expressions \((x_2 - x_1)^2\) and \((y_2 - y_1)^2\) represent algebraic forms that simplify the comparison of distances across a plane.
• Squaring these expressions achieves two things: it removes the influence of any negative signs since a distance is always positive, and it prepares the base for calculating a square root, ensuring we're dealing with the correct length.

In this case:

• \(3^2 = 9\) and \(4^2 = 16\)
• Addition of these squares, followed by taking the square root, gives us a logical and clean path from coordinates to true mathematical distance.

Understanding algebraic expressions in such operations emphasizes the role of algebra as the linking language between numerical calculations and geometric understanding.