Problem 78 - Algebra A Combined Function

Question

Find the distance between each pair of points. Give an exact distance and a three-decimal-place approximation. (2,3) and (14,8)

Step-by-Step Solution

Answer

Distance = 13, both exact and approximated to three decimals as 13.000.

Step 1: Understand the distance formula

To find the distance between two points \(x_1, y_1\) and \(x_2, y_2\), we use the distance formula: \(\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)This formula comes from the Pythagorean theorem, treating the distance as the hypotenuse of a right triangle formed by differences in x and y coordinates.

Step 2: Substitute the coordinates into the formula

Substitute the given points (2, 3) and (14, 8) into the distance formula: \(\text{Distance} = \sqrt{(14 - 2)^2 + (8 - 3)^2}\)

Step 3: Calculate the differences

Calculate the differences: \(14 - 2\) which equals 12, and \(8 - 3\) which equals 5.

Step 4: Square the differences

Square each of the differences calculated in the previous step: \(12^2 = 144\) and \(5^2 = 25\).

Step 5: Add the squares of the differences

Add the squares of the differences: \(144 + 25 = 169\)

Step 6: Find the square root of the sum

The distance is the square root of the sum from the previous step: \(\sqrt{169} = 13\). This is the exact distance.

Step 7: Approximate the distance to three decimal places

Since the exact distance found is an integer (13), rounding to three decimal places doesn't change it: the approximate distance is still 13.000.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a mathematical system that uses ordered pairs (coordinates) to determine positions on a plane. This approach allows us to use algebraic methods to solve geometric problems. By plotting points on the coordinate plane using two values - x for horizontal positioning and y for vertical positioning, we can visually interpret and calculate geometric properties, such as distances and slopes.

In the context of the distance calculation between two points, the coordinates (x, y) work like an address system that locates each point precisely on the plane. Understanding this concept helps you effectively apply mathematical formulas, such as the distance formula, to determine the distances between points on the plane defined by their specific coordinates.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, used to calculate the length of the sides in a right triangle. The theorem states: 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 other two sides.

Mathematically, it's expressed as:

• \( a^2 + b^2 = c^2 \)

When solving distance problems in coordinate geometry, imagine creating a right triangle where the line connecting two points forms the hypotenuse. By calculating the differences in x and y coordinates, you essentially divide the distance problem into two simpler perpendicular segments, which you can analyze using the Pythagorean Theorem.

This understanding allows for the translation of a geometric problem into a manageable algebraic exercise, using the Pythagorean Theorem to confidently derive the exact length of the hypotenuse, which corresponds to the distance between the two points.

Exact Distance Calculation

Exact distance calculation involves using coordinate geometry and the distance formula derived from the Pythagorean Theorem. With two points on the coordinate plane, represented as

• \((x_1, y_1)\) and \((x_2, y_2)\),

we calculate the exact distance as follows:

• The difference between their `x` values \((x_2 - x_1)\)
• And their `y` values \((y_2 - y_1)\)

Next, square these differences to eliminate any negative signs. Adding these squared differences gives the sum, from which the square root is found to give the precise distance:

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

This exact calculation is crucial in geometry and real-world applications where precision is vital, such as in engineering and physics. By following these steps, the distance is accurately determined, ensuring that no measurement is left to estimation.

Square Root Approximation

Square root approximation is sometimes necessary when the exact value involves irrational numbers (numbers that cannot be precisely expressed as a simple fraction). Though in our earlier calculation, the exact distance was a perfect square (13), often, you'll encounter distances that require approximation.

When approximate values are needed, especially to a specific number of decimal places, this is typically done using mathematical or digital tools, which help refine the approximate square root to a desired precision. The process usually includes:

• Identifying the nearby integers (e.g., estimating \(\sqrt{150}\) between \( \sqrt{144} = 12\) and \(\sqrt{169} = 13\))
• Using a calculator or algorithmic function to pinpoint closer decimal placements
• Rounding to the required decimal places for precision

In practice, rounding to three decimal places maintains an accuracy suitable for most practical geometry applications. These approximations are essential in everyday calculations, where an exact whole number is rare or impossible to maintain.