Problem 12 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 9-14: (a) Find the distance \(d(A, B)\) between \(A\) and \(B\). (b) Find the midpoint of the segment \(A B\). $$ A(6,2), \quad B(6,-2) $$

Step-by-Step Solution

Answer

The distance between points A and B is 4, and the midpoint is (6, 0).

Step 1: Use the Distance Formula

The distance formula between two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is \(d(A, B) = \sqrt{{(x_2 - x_1)}^2 + {(y_2 - y_1)}^2}\). Substitute the coordinates for points A and B: \(x_1 = 6, y_1 = 2, x_2 = 6, y_2 = -2\).

Step 2: Substitute and Simplify

Substituting the values into the distance formula gives: \[d(A, B) = \sqrt{{(6 - 6)}^2 + {(-2 - 2)}^2}\]. Simplifying inside the square root results in \[d(A, B) = \sqrt{0 + 16} = \sqrt{16} = 4\].

Step 3: Use the Midpoint Formula

The midpoint formula for segment AB with endpoints \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is \(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). Substitute \(x_1 = 6, y_1 = 2, x_2 = 6, y_2 = -2\) into the formula.

Step 4: Calculate the Midpoint

The midpoint is calculated as \(M = \left(\frac{6 + 6}{2}, \frac{2 + (-2)}{2}\right) = (6, 0)\).

Key Concepts

Distance Formula

In coordinate geometry, the distance between two points in a plane can be calculated using the distance formula. The formula is derived from the Pythagorean theorem and it provides a way to find the straight-line distance between two points, say point \(A(x_1, y_1)\) and point \(B(x_2, y_2)\), in the Cartesian plane. Here's how it works:

• The formula is: \(d(A, B) = \sqrt{{(x_2 - x_1)}^2 + {(y_2 - y_1)}^2}\).
• This equation accounts for both the horizontal and vertical differences between the two points.
• By squaring each difference, you effectively measure those differences without concern for direction, emphasizing the actual distance.

Using our example, points \(A(6,2)\) and \(B(6,-2)\) were plugged into this formula. The horizontal difference \((6-6)\) is zero because both points have the same x-coordinate.
This results in a purely vertical distance calculation: \(\sqrt{0 + 16} = 4\). It clearly shows how even without horizontal change, distance can still be calculated accurately when there is a vertical change.
Understanding the distance formula is essential in many real-world problems, such as finding how far apart two locations are on a map.

Midpoint Formula

The midpoint formula in coordinate geometry is a method for finding the exact middle point of a line segment connecting two points. This is especially useful for bisecting a line segment or determining a balance point on a graph.

• The formula to find the midpoint \(M\) between two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is: \(M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\).
• It essentially averages the x-coordinates and the y-coordinates of the points \(A\) and \(B\).
• This formula is symmetric, meaning it does not depend on the order of the points.

For example, with points \(A(6,2)\) and \(B(6,-2)\), when we apply the midpoint formula, the result is \((6, 0)\).
Here, the x-values didn't change when averaged, because they are the same; \(6\).
The y-values average to zero, reflecting their symmetrical positioning around the x-axis. Recognizing and using the midpoint formula allows us to find this middle point, useful in dividing spaces or determining points in navigation.

Cartesian Plane

The Cartesian plane, named after René Descartes, is a fundamental part of coordinate geometry. It is a two-dimensional plane defined by a horizontal and vertical axis intersecting at a point called the origin.

• The horizontal axis is typically labeled the x-axis, and the vertical as the y-axis.
• Every point on the plane can be represented as a coordinate \((x, y)\).
• The origin lies at \((0, 0)\), where the x-axis and y-axis meet.

This plane allows us to graphically represent mathematical equations and problems. In the example with points \(A(6, 2)\) and \(B(6, -2)\), these points are located on this plane where their x-coordinates are the same.
This means they are aligned vertically. The Cartesian plane provides a visual representation that can help simplify complex numerical relationships and offer insights not easily gleaned from equations alone.
It is a powerful tool for visual learners and enhances understanding of mathematical concepts in both educational and practical scenarios.