Problem 14 - 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. $$ (5,0),(0,6) $$

Step-by-Step Solution

Answer

Distance: \( \sqrt{61} \), Midpoint: \( \left( \frac{5}{2}, 3 \right) \).

Step 1: Plot the Points

To plot the points (5, 0) and (0, 6) on a coordinate plane, locate the first point (5, 0) by moving 5 units along the x-axis and 0 units along the y-axis. For the second point (0, 6), move 0 units along the x-axis and 6 units up along the y-axis.

Step 2: Use the Distance Formula

The distance formula is used to find the distance between two points, which is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Substituting the given points (5, 0) and (0, 6) into the formula, we have \( d = \sqrt{(0 - 5)^2 + (6 - 0)^2} = \sqrt{(-5)^2 + (6)^2} = \sqrt{25 + 36} = \sqrt{61} \).

Step 3: Find the Midpoint

The midpoint of a segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). For the points (5, 0) and (0, 6), the midpoint is \( M = \left( \frac{5 + 0}{2}, \frac{0 + 6}{2} \right) = \left( \frac{5}{2}, 3 \right) \).

Key Concepts

Distance Formula

The distance formula in coordinate geometry is an essential tool for finding the distance between two points on a coordinate plane. Imagine two points, like the ends of a line, and we want to know how far apart they are. This is where the distance formula comes into play. To use the distance formula, you need the coordinates of the two points you wish to measure. Let's say you have two points \( (x_1, y_1) \) and \( (x_2, y_2) \).

• Step 1: Subtract the x-coordinates: \( x_2 - x_1 \)
• Step 2: Subtract the y-coordinates: \( y_2 - y_1 \)
• Step 3: Square both results: \( (x_2 - x_1)^2 \) and \( (y_2 - y_1)^2 \)
• Step 4: Add these squares: \( (x_2 - x_1)^2 + (y_2 - y_1)^2 \)
• Step 5: Finally, take the square root: \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

In our example with points \( (5, 0) \) and \( (0, 6) \), you follow these steps and find that the distance between them is \( \sqrt{61} \). By practicing these steps, you'll quickly become proficient at calculating distances on a coordinate plane.

Midpoint Formula

The midpoint formula is used when you want to find a point exactly halfway between two other points. It's like finding the center point of a segment. The formula is easy to remember and apply. Given points \( (x_1, y_1) \) and \( (x_2, y_2) \), the midpoint \( M \) has coordinates:

• For the x-coordinate: \( \frac{x_1 + x_2}{2} \) - Take the average of the x-values.
• For the y-coordinate: \( \frac{y_1 + y_2}{2} \) - Take the average of the y-values.

By averaging the x-coordinates and the y-coordinates of the given points, you find a new point that sits right in the middle. For our points \( (5, 0) \) and \( (0, 6) \), following the formula gives a midpoint of \( (\frac{5}{2}, 3) \). This midpoint represents a balanced location along the segment connecting the two original points.

Plotting Points

Plotting points is one of the basic yet critical skills in coordinate geometry. It involves placing points on a graph based on their coordinates. Each point is defined by a pair \( (x, y) \), where \( x \) is the horizontal position and \( y \) is the vertical position. Follow these simple steps to plot any given point:

• Start from the origin \( (0,0) \), the central point where the x-axis and y-axis meet.
• Move horizontally along the x-axis by the value of the x-coordinate. If the x-coordinate is positive, move to the right; if negative, move to the left.
• Next, move vertically up or down based on the y-coordinate. If the y-coordinate is positive, move up; if negative, move down.

By following these steps, you can accurately place any point on a grid. In our specific exercise, points \( (5, 0) \) and \( (0, 6) \) are plotted by moving 5 units right from the origin for the first point, and 6 units up for the second point. Mastering the art of plotting will help you visualize problems and find solutions with ease.