Problem 77 - Algebra A Combined Function

Question

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. (5,1) and (8,5)

Step-by-Step Solution

Answer

Distance is 5.000.

Step 1: Identify the Points

First, identify the coordinates of the two points given in the problem. The points are \((5,1)\) and \((8,5)\). Here, \((x_1, y_1) = (5, 1)\) and \((x_2, y_2) = (8, 5)\).

Step 2: Understand the Distance Formula

The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is found using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Step 3: Substitute the Coordinates

Substitute the coordinates of the two points into the distance formula. This gives: \[ d = \sqrt{(8 - 5)^2 + (5 - 1)^2} \]

Step 4: Calculate the Distances

Calculate the differences and their squares: \((8 - 5)^2 = 3^2 = 9\) and \((5 - 1)^2 = 4^2 = 16\).

Step 5: Sum and Simplify

Add these squared differences: \[ d = \sqrt{9 + 16} = \sqrt{25} \]

Step 6: Compute the Exact Distance

The square root of 25 is 5, thus the exact distance is \(d = 5\).

Step 7: Approximate the Distance

In this case, since the distance \(d = 5\) is already an exact integer, the approximation to three decimal places is simply 5.000.

Key Concepts

Exact Distance Calculation

The exact distance calculation is a precise method used to determine the length of the straight line between two points in a coordinate plane. The formula used for this is the well-known distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] This formula essentially comes from the Pythagorean theorem. By calculating the horizontal and vertical differences between the points and squaring them, we find the lengths of the sides of a right triangle whose hypotenuse is the distance we need to calculate. To find the exact distance:

• Subtract the x-coordinates and square the result.
• Subtract the y-coordinates and square the result.
• Add these two squared values together.
• Take the square root of the sum to find the exact distance.

In the example of points \((5,1)\) and \((8,5)\), the exact distance is calculated as:\[ d = \sqrt{(8 - 5)^2 + (5 - 1)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \] This method gives us the precise distance without any approximation.

Approximation

Approximation involves finding a value close enough to the exact number to be useful for practical purposes. It is often used to simplify complex numbers or expressions. Generally, this is useful when dealing with irrational numbers that cannot be expressed exactly as fractions.In our case, though the exact distance was calculated as 5 (a simple number), expressing this as a three-decimal-place approximation could be useful in situations requiring standardized numeric formats.
In most calculations, you'll handle decimal numbers that need rounding to suit a specific precision level. For instance:

• If the number were not as simple as 5, and its approximate value was 5.56789, we'd round it to 5.568 for simpler representation.
• If our exact answer was an irrational number like \( \sqrt{2} \), we'd find a decimal approximation like 1.414.
• In our specific exercise, since the exact integer was 5, the three-decimal approximation remains 5.000.

Approximations help streamline calculations and make communication of results easier in practical applications.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, plays a critical role in understanding and solving geometric problems by using a coordinate system. It combines algebra and geometry, allowing for the simplistic algebraic manipulation of geometric figures. Points, lines, and shapes are represented using coordinates, which simplifies calculations related to distance, area, and other geometric properties. Key aspects in coordinate geometry include:

• Points, defined by their coordinates \((x, y)\).
• Lines, described using equations like \(y = mx + b\).
• Shapes, which can be analyzed for their geometric properties by using their coordinates.

By applying coordinate systems, you can easily solve problems with less visual ambiguity. For instance, finding the distance between two points (\((x_1, y_1)\) and \((x_2, y_2)\)) becomes a straightforward algebra problem using the distance formula.In summary, coordinate geometry provides powerful tools for solving problems, making complex spatial relationships simpler to understand and compute.