Problem 54 - Elementary and Intermediate Algebra

Question

Find the exact distance between each pair of points. \((4,7),(-4,-5)\)

Step-by-Step Solution

Answer

The exact distance is \(4\sqrt{13}\).

Step 1: Identify the coordinates of the points

We have two points: Point A with coordinates \((x_1, y_1) = (4, 7)\) and Point B with coordinates \((x_2, y_2) = (-4, -5)\).

Step 2: Apply 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 values from Step 1 into the distance formula: \[d = \sqrt{(-4 - 4)^2 + (-5 - 7)^2}\] which simplifies to: \[d = \sqrt{(-8)^2 + (-12)^2}\]

Step 4: Calculate the squares of the differences

Calculate the squares inside the square root: \[(-8)^2 = 64 \quad \text{and} \quad (-12)^2 = 144\].

Step 5: Add the calculated squares

Add the squared values: \[64 + 144 = 208\].

Step 6: Take the square root

Take the square root of the sum to find the exact distance: \[d = \sqrt{208} = \sqrt{16 \times 13} = 4\sqrt{13}\].

Step 7: Final Result

The exact distance between the points \((4,7)\) and \((-4,-5)\) is \(4\sqrt{13}\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This method allows us to describe geometric shapes numerically and derive algebraic equations to represent lines, curves, and figures.
In coordinate geometry, each point is specified by a pair of numerical coordinates. These coordinates, usually written as \(x, y\), represent the point's position on the Cartesian plane. This plane is divided into four quadrants by the x-axis (horizontal) and the y-axis (vertical).
Working within this system, you can easily calculate distances, gradients, and areas. By converting geometric problems into algebraic equations, coordinate geometry provides a powerful tool for mathematical problem solving.

Distance Between Points

One of the fundamental concepts in coordinate geometry is finding the distance between two points. The distance formula is derived from the Pythagorean theorem and applies to any two points in a plane.
The formula to find the distance \(d\) between two points with coordinates \(x_1, y_1\) and \(x_2, y_2\) is:

• \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

Breaking this down, the formula calculates the horizontal and vertical distances between the points, squares them, and finds their sum. The square root of this sum gives the straight-line distance between the points.
For example, for points \(4, 7\) and \(-4, -5\), the distance is calculated as \([(-4 - 4)^2 + (-5 - 7)^2]^{1/2}\), simplifying to \([64 + 144]^{1/2} = \sqrt{208} = 4\sqrt{13}\). This shows the power of coordinate geometry in solving problems easily with a standardized method.

Mathematical Problem Solving

Mathematical problem solving involves a variety of skills and habits of thought that enable tackling diverse problems.
To solve problems like finding the distance between points, it's crucial to:

• Understand the problem and identify what is given.
• Select an appropriate strategy or formula. For example, using the distance formula for coordinate geometric problems.
• Execute calculations carefully and accurately.
• Reflect on the solution to ensure its reasonableness and clarity.

This structured approach not only simplifies solving geometric problems but also builds proficiency in tackling other mathematical challenges. By systematically applying these strategies, you gain confidence and skill in problem solving across different areas of mathematics.