Problem 12 - Algebra and Trigonometry

Question

\(7-14\) A pair of points is graphed. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the midpoint of the segment that joins them. $$ (-1,6),(-1,-3) $$

Step-by-Step Solution

Answer

Points are vertically aligned, distance is 9 units, midpoint is (-1, 1.5).

Step 1: Plot the Points

Identify the given points: Point A is (-1, 6) and Point B is (-1, -3). Plot these points on a coordinate plane. Locate the x-coordinate (-1) on the horizontal axis, then move up to the y-coordinate (6) for Point A and down to (-3) for Point B.

Step 2: Calculate the Distance

Use the distance formula to find the distance between two points: \( \text{Distance} = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \). Since both x-coordinates are (-1), (x_2-x_1) = 0. Thus, calculate \( \sqrt{0 + (6 - (-3))^2} = \sqrt{9^2} = 9 \). The distance is 9 units.

Step 3: Find the Midpoint

Use the midpoint formula to find the midpoint of a segment: \( \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right) \). Substitute the given points to get the midpoint: \( \left( \frac{-1 + (-1)}{2}, \frac{6 + (-3)}{2} \right) = \left( -1, \frac{3}{2} \right) \). The midpoint is (-1, 1.5).

Key Concepts

Distance Formula in Coordinate Geometry

In coordinate geometry, the distance formula is a very handy tool. It helps us find the exact distance between two points on a coordinate plane. This formula is essentially derived from the Pythagorean theorem.

Here’s how the distance formula works: given two points, (A: \((x_1, y_1)\) and B: \((x_2, y_2)\)), the formula to calculate the distance \(d\) between them is:

• \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

By looking at the formula, you can see it's important to square the differences between the x-coordinates and the y-coordinates. Then, we add these squared differences. Finally, take the square root of this sum to find the distance. This might sound complex at first, but once you plug in the numbers, it becomes clearer.

For our specific example of points \((-1, 6)\) and \((-1, -3)\), notice that the x-coordinates are identical. So, the formula simplifies a bit: the difference in x is zero. Therefore, the distance solely depends on the y-coordinates: \(6 - (-3)\), which is \(9\). This step simplifies to \(\sqrt{9^2} = 9\). Thus, the distance is 9 units.

Understanding the Midpoint Formula

The midpoint formula helps us find the middle point of a line segment joining two given points. This is useful when you need to know the exact center point. The midpoint is calculated by finding the average of the x-coordinates and the y-coordinates separately for the two points.

Given two points, (A: \((x_1, y_1)\) and B: \((x_2, y_2)\)), the formula to find the midpoint \(M\) is:

• \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)

It is as if you are simply averaging the x-values and the y-values independently to find the exact center between these two points.

For the points \((-1, 6)\) and \((-1, -3)\), the x-coordinates being the same simplifies the calculation. Substitute these into the formula: \( \left( \frac{-1 + (-1)}{2}, \frac{6 + (-3)}{2} \right) = (-1, 1.5)\). Hence, the midpoint lies at \((-1, 1.5)\). This tells us that, visually, the midpoint of our vertical line segment between these points lies half-way along the y-axis at \(1.5\).

Plotting Points on the Coordinate Plane

Plotting points on a coordinate plane sets the stage for utilizing both the distance and midpoint formulas effectively. When you have a pair of coordinates like \((-1, 6)\) and \((-1, -3)\), the process begins by identifying the x and y values for each point. The first number in each pair is the x-coordinate, and the second is the y-coordinate.

To plot a point:

• Locate the x-coordinate on the horizontal axis.
• From there, move vertically to the appropriate y-coordinate.

For example, for the point \((-1, 6)\), first locate \(-1\) on the x-axis. Then move up to \(6\) on the y-axis to plot the point. Similarly, for \((-1, -3)\), find \(-1\) on the x-axis and move down to \(-3\) on the y-axis.

This simple process of plotting allows you to visually see the points and the line segment connecting them. Understanding where these points lie in relation to each other is crucial for applying further analysis such as measuring distance or finding midpoints.