Problem 49 - Algebra 2

Question

Find the distance between each pair of points with the given coordinates. $$ (4,-1),(-2,7) $$

Step-by-Step Solution

Answer

The distance between the two points is 10 units.

Step 1: Identify the coordinates

First, identify and label the coordinates of the two points given. The first point is \((4, -1)\) and the second point is \((-2, 7)\).

Step 2: Use the distance formula

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

Step 3: Substitute the coordinates into the formula

Substitute the coordinates into the distance formula. Here, \(x_1 = 4\), \(y_1 = -1\), \(x_2 = -2\), and \(y_2 = 7\):\[ d = \sqrt{((-2) - 4)^2 + (7 - (-1))^2} \]

Step 4: Simplify inside the parentheses

Simplify the expressions inside the parentheses:\[ d = \sqrt{(-6)^2 + (8)^2} \]

Step 5: Calculate the squares

Calculate the square of each term: \[ (-6)^2 = 36 \] \[ (8)^2 = 64 \] Therefore, \[ d = \sqrt{36 + 64} \]

Step 6: Sum and take the square root

Add the results and take the square root: \[ d = \sqrt{100} \] This simplifies to: \[ d = 10 \]

Step 7: Conclusion

The distance between the points \((4, -1)\) and \((-2, 7)\) is 10 units.

Key Concepts

Coordinate Geometry

Coordinate Geometry is a branch of geometry where we use algebraic expressions and coordinate systems to represent geometric figures. A key part of coordinate geometry is the use of coordinates, usually represented as pairs of numbers. Each pair provides a precise location or point on a plane. In our exercise, these pairs are

• Point 1: \((4, -1)\)
• Point 2: \((-2, 7)\)

The values represent locations on a two-dimensional plane, called a Cartesian plane. The first number in each pair is the x-coordinate (horizontal position), and the second number is the y-coordinate (vertical position). This system allows us to perform operations like finding distances between points or angles between lines, all using algebraic formulas.

Euclidean Distance

Euclidean Distance is the term used in mathematics to describe the "ordinary" straight-line distance between two points in a Euclidean space, which can be thought of as our regular 2D or 3D space. It is named after the ancient Greek mathematician Euclid.To calculate this distance between two points, we use the Distance Formula. For two points with coordinates \((x_1, y_1)\) and \((x_2, y_2)\), the formula is:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula is a direct result of the Pythagorean theorem, which applies to the right triangle formed by the vertical and horizontal distances between the points. In our example, we find the distance between \((4, -1)\) and \((-2, 7)\) by substituting the coordinates into the formula, step by step simplifying:

• \((-2) - 4 = -6\)
• \(7 - (-1) = 8\)
• \((-6)^2 = 36, \quad (8)^2 = 64\)
• \(36 + 64 = 100\)
• \(\sqrt{100} = 10\)

Thus, the Euclidean distance between these points is 10 units.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It is about finding the unknown or putting real-life variables into equations and then solving them. In the context of our exercise, algebra is used to plug the coordinates into the Distance Formula to find the Euclidean distance. The manipulation involves basic operations: addition, subtraction, multiplication, and finding square roots. Here’s how algebra plays a role in our problem:

• First, we subtract the x-coordinates and the y-coordinates separately.
• Then, we square those differences to eliminate negative values and emphasize the magnitude.
• Next, we sum the squares.
• Finally, we take the square root of this sum to return to the original units of measurement (distance).

Each step requires logical reasoning, relying on algebraic principles to solve real-world problems like measuring distances accurately on a coordinate plane.