Problem 47 - Elementary and Intermediate Algebra

Question

Find the exact distance between each pair of points. \((6,8),(12,16)\)

Step-by-Step Solution

Answer

The distance between the points is 10.

Step 1: Identify the Formula

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

Step 2: Assign Point Values

Assign the given coordinates to the formula. Let \(x_1, y_1\) be \(6, 8\) and \(x_2, y_2\) be \(12, 16\).

Step 3: Substitute Values Into Formula

Insert the values into the distance formula: \[ d = \sqrt{(12 - 6)^2 + (16 - 8)^2} \].

Step 4: Calculate Differences and Squares

Calculate the differences in x and y coordinates: \(12 - 6 = 6\) and \(16 - 8 = 8\). Then square these differences: \(6^2 = 36\) and \(8^2 = 64\).

Step 5: Sum the Squares

Add the squared values from the previous step: \(36 + 64 = 100\).

Step 6: Find the Square Root

Calculate the square root of the sum: \[\sqrt{100} = 10\].

Step 7: Conclusion

The exact distance between the points \(6, 8\) and \(12, 16\) is 10.

Key Concepts

Understanding Coordinate Geometry

In the world of mathematics, coordinate geometry is a crucial concept. It's a branch of geometry where we use numerical coordinates to represent points on a plane. These coordinates are usually written as pairs: \(x, y\). The first number, \(x\), represents the position on the horizontal axis, while the second number, \(y\), indicates the position on the vertical axis. In simpler terms, coordinate geometry allows us to describe any point precisely on a 2-dimensional surface using numbers.
This mathematical tool helps us in many ways:

• It simplifies the way we represent geometric shapes and figures by using numbers.
• It provides a foundation for understanding more complex topics such as vectors and matrices.
• It enables us to solve problems involving locations, paths, and understand spatial relationships better.

In this exercise, the points \(6, 8\) and \(12, 16\) are represented on a 2D plane. Our task is to find the distance between them, which we can achieve using the principles of coordinate geometry.

Calculating Distances in Coordinates

When we talk about measuring distances between two points in coordinate geometry, the distance formula is key. Derived from the Pythagorean theorem, it helps us calculate how far apart two points are, considering both horizontal and vertical differences. The formula is \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \], where \(d\) represents the distance, and \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of the two points.
Let’s break down the process of distance calculation:

• Step 1: Identify the coordinates of the points you want to measure.
• Step 2: Substitute these values into the distance formula.
• Step 3: Calculate the differences in coordinates, both horizontally and vertically.
• Step 4: Square these differences.
• Step 5: Add the squared values to find the sum.
• Step 6: Take the square root of the sum to find the final distance.

By using these steps, we ensure accurate and effective measuring between any two points on a grid, such as from \(6, 8\) to \(12, 16\) in our task.

Exploring Algebraic Expressions

Algebraic expressions play a significant role in the journey from understanding coordinate geometry to calculating distances. An algebraic expression is a mathematical phrase that can include numbers, variables, and operation signs. In the context of our exercise, we manipulate such expressions to find the distance between two points.
Let's look at how this applies here:

• Variables as Coordinates: When points are given, like \( (6, 8) \) and \( (12, 16) \), they become the variables in our distance formula.
• Performing Operations: Subtraction is the first operation we use in \(x_2 - x_1\) and \(y_2 - y_1\). Following that, squaring involves multiplying these results by themselves.
• Simplifying Results: Once you add these squared results, you simplify the expression using square roots to find the distance.

Algebra helps us to methodically tackle each part of a problem. By applying algebraic operations, we manage to transform abstract numbers into tangible results, like determining the distance between two specific points in this exercise. Through these expressions, we harness the logic and precision of algebra to solve problems.