Problem 79 - Intermediate Algebra

Question

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. See Example 6 $$ (-3,2) \text { and }(1,-3) $$

Step-by-Step Solution

Answer

The exact distance is \(\sqrt{41}\), approximately 6.403.

Step 1: Understand the Distance Formula

To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in the coordinate plane, we use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] This formula is derived from the Pythagorean Theorem, where the distance between the points forms the hypotenuse of a right triangle.

Step 2: Identify the Coordinates

For the given points, \((-3, 2)\) and \((1, -3)\), identify the coordinates: - \(x_1 = -3\), \(y_1 = 2\) - \(x_2 = 1\), \(y_2 = -3\)

Step 3: Substitute into the Distance Formula

Substitute the coordinates into the distance formula: \[ d = \sqrt{(1 - (-3))^2 + (-3 - 2)^2} \] This simplifies to: \[ d = \sqrt{(1 + 3)^2 + (-5)^2} \]

Step 4: Calculate the Distances Inside the Formula

Calculate inside the square root: \[ d = \sqrt{(4)^2 + (-5)^2} \] This becomes: \[ d = \sqrt{16 + 25} \]

Step 5: Simplify the Expression

Add the numbers inside the square root: \[ d = \sqrt{41} \] This is the exact distance between the points.

Step 6: Find the Decimal Approximation

Using a calculator, \[ \sqrt{41} \approx 6.403 \] This is the distance to three decimal places.

Key Concepts

Coordinate Geometry

Coordinate geometry is a powerful tool in mathematics that allows us to describe the positions and relationships between points on a plane using a set of numerical coordinates. These coordinates are typically defined as

• an x-coordinate, which measures the horizontal Distance from the origin
• a y-coordinate, which measures the vertical Distance from the origin

By using these two values, we can precisely locate any point on a 2D coordinate system. In the context of finding distances, coordinate geometry helps us calculate how far apart two points are based on their respective coordinates.
Understanding this allows us to move into using the Distance Formula, deeply intertwined with these coordinate concepts.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in mathematics, particularly in geometry. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed with the equation:\[ c^2 = a^2 + b^2 \]In the context of the Distance Formula, imagine the two points we are interested in as forming the hypotenuse of a right triangle on the coordinate plane. The vertical distance between the points forms one leg of the triangle, and the horizontal distance forms the other.
This geometric visualization is how we derive the Distance Formula, providing a bridge between algebra and geometry through this classic theorem.

Decimal Approximation

Decimal approximation refers to the practice of rounding numbers to a certain number of decimal places to make them easier to work with. When dealing with irrational numbers, like the square root of 41 in this exercise, the exact value can be challenging to handle in practical situations.
By finding a decimal approximation of \( \sqrt{41} \), we transition from an exact form (\( \sqrt{41} \)), which remains abstract, to approximately 6.403. This provides a more manageable number that can be used in further calculations or when communicating measurements, crucial for practical applications and real-world contexts.

• This rounding process ensures clarity and comprehensibility without losing essential information about the value.

Mathematical Calculation

Mathematical calculations involve performing operations to arrive at a numerical result. In our exercise, we started by substituting the given coordinates into the Distance Formula. This step-by-step process is crucial:

• Identify the coordinates of the points.
• Use these coordinates to substitute into the Distance Formula:
• Simplify the equation inside the square root.
• Find both the exact and approximate values.

Performing these calculations accurately allows us to determine the exact distance, \( \sqrt{41} \), and the decimal approximation, 6.403.
Approaching calculations methodically ensures that each stage of the problem is tackled thoroughly, helping to reduce errors and increase confidence in the result.