Problem 48 - Algebra 2

Question

Find the distance between each pair of points with the given coordinates. $$ (7,3),(-5,8) $$

Step-by-Step Solution

Answer

The distance is 13.

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: \( ext{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Step 2: Substitute the coordinates into the formula

Assign \(x_1 = 7\), \(y_1 = 3\), \(x_2 = -5\), and \(y_2 = 8\) to the distance formula. Replace the variables with these values: \( ext{Distance} = \sqrt{((-5) - 7)^2 + (8 - 3)^2}\).

Step 3: Simplify inside the square root

First, calculate \((-5) - 7 = -12\) and \(8 - 3 = 5\). Substitute these results back into the formula: \( ext{Distance} = \sqrt{(-12)^2 + 5^2}\).

Step 4: Perform the squaring operations

Calculate \((-12)^2 = 144\) and \(5^2 = 25\). Substitute back into the formula: \( ext{Distance} = \sqrt{144 + 25}\).

Step 5: Add the results inside the square root

Add 144 and 25 to get \(169\). Thus, the formula simplifies to \( ext{Distance} = \sqrt{169}\).

Step 6: Calculate the square root

Find the square root of 169, which is 13. So, \( ext{Distance} = 13\).

Key Concepts

Square Root

The square root is a mathematical operation that finds a number which, when multiplied by itself, will equal the given value. For example, the square root of 169 is 13, because 13 multiplied by 13 equals 169. In the context of the distance formula, calculating the square root is the final step to determine the linear distance between two points.
The square root symbol is denoted by \( \sqrt{} \), and it simplifies the result from squaring calculations and additions within the formula. It's essential to remember that the square root returns the principal (or non-negative) root of a number.
Understanding how to calculate square roots is fundamental in moving from squared differences to an actual measured distance, making it a crucial step in coordinate geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses algebraic equations to describe geometric principles. It allows us to find the distance between points using their coordinates. The distance formula comes from the Pythagorean theorem applied in a coordinate plane.
Each point in coordinate geometry is represented by an ordered pair \((x, y)\). With this method, we can calculate distances, slopes, and even the equations of lines. Coordinate geometry bridges between algebra and geometry, providing a powerful tool for understanding space and shapes.

• Each pair of coordinates on a plane is unique.
• Distances can be calculated in any direction or axis on the plane.
• It applies to both positive and negative coordinate values.

Coordinate geometry simplifies the visualization of geometric concepts by placing them in a numerical format.

Distance Calculation

Distance calculation between two points involves using their coordinates and the distance formula. The given formula is \( \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). This computes the straight-line distance, or Euclidean distance, between the two points in a plane.
In this formula:

• \(x_1\) and \(y_1\) are the coordinates of the first point.
• \(x_2\) and \(y_2\) are the coordinates of the second point.
• The differences \((x_2 - x_1)\) and \((y_2 - y_1)\) measure how far apart the points are in the horizontal and vertical directions.

By squaring these differences, each axis's separation contributes positively to the total distance. Adding these squares and finding their square root yields the exact distance.
This method is not limited by the orientation of points, making it a universal way to calculate distances in a coordinate system.