Problem 14 - College Algebra with Modeling and Visualization

Question

Find the exact distance between the two points. Where appropriate, also give approximate results to the nearest hundredth. $$ (0,-3),(12,-8) $$

Step-by-Step Solution

Answer

The exact distance is 13 units.

Step 1: Identify the Formula

To find the distance between two points \(x_1, y_1\) and \(x_2, y_2\) on a 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.

Step 2: Substitute the Values

Substitute the given points \(0, -3\) and \(12, -8\) into the distance formula. - Here, \(x_1 = 0, y_1 = -3, x_2 = 12, y_2 = -8\).- Therefore, the formula becomes: \[ d = \sqrt{(12 - 0)^2 + (-8 + 3)^2} \]

Step 3: Simplify the Arithmetic

Compute inside the parentheses first:- Calculate \(12 - 0\), which equals 12.- Calculate \(-8 + 3\), which equals -5.

Step 4: Square the Results

Next, we find the square of the results from the previous step:- \(12^2 = 144\)- \((-5)^2 = 25\)

Step 5: Add the Squares

Add the squared numbers: - Add 144 and 25 to get 169.

Step 6: Take the Square Root

To find the distance, take the square root of the final sum:- \(\sqrt{169} = 13\)- Therefore, the exact distance is 13.

Step 7: Approximation (No Need for Further Rounding)

The distance calculated as 13 is an exact integer, hence it doesn't require approximation to the nearest hundredth.

Key Concepts

Coordinate Geometry

Coordinate Geometry is an essential branch of geometry that deals with points, lines, and shapes on the coordinate plane. It allows us to precisely determine positions and compute geometric properties using algebraic equations.
In a coordinate plane, each point is defined by an ordered pair \(x, y\). This concept simplifies complex geometry problems by converting them into algebraic expressions, making it easier to understand and solve them.
For instance:

• A point \(0, -3\) is located at the origin along the X-axis at zero and three units below along the Y-axis.
• A point \(12, -8\) means moving twelve units right and eight units down from the origin.

Coordinate geometry is foundational for finding distances between points, as seen in the problem above using the Distance Formula. This formula efficiently computes the exact length between any two points based on their coordinates, converting geometric concepts into manageable calculations.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry expressing the relationship between the sides of a right triangle. The theorem states that for a triangle with sides labeled as 'a', 'b', and hypotenuse 'c', the equation \(a^2 + b^2 = c^2\) holds true.
This theorem is the basis for the Distance Formula. By creating a right triangle from two points on a coordinate plane, where the legs are the differences in the x and y coordinates, we can find the hypotenuse, representing the distance between the points:

• Horizontal Distance: Calculated as \(x_2 - x_1\)
• Vertical Distance: Calculated as \(y_2 - y_1\)

The Distance Formula \(\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]\) directly embodies the Pythagorean Theorem, using the squares and square root to determine the distance as if you were solving for the hypotenuse of a right triangle.

Mathematical Calculations

Mathematical Calculations are indispensable in solving Distance Formula problems. Each operation demands careful attention to detail to ensure accuracy.
Let's review the necessary calculations from our step-by-step solution:

• Identify the Coordinates: Clearly define the x and y variables for both points, \((x_1, y_1)\) and \((x_2, y_2)\).
• Subtract and Simplify: Subtract corresponding coordinates to get differences: \(x_2 - x_1\) and \(y_2 - y_1\).
• Square the Differences: Apply squaring to each difference to achieve positive values, which aligns with geometric distance's non-negative property.
• Add Results: Sum the squared values, a decisive step to find the total squared distance.
• Take the Square Root: Finally, find the square root of the sum to determine the exact distance between the two points.

Executing these calculations accurately results in reliable solutions, allowing us to confidently express the exact distance as '13' in this context. This process highlights the methodical nature of mathematical problem-solving, providing clarity and precision.