Problem 53 - Precalculus with Limits

Question

In Exercises 47-56, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \( (\frac{1}{2}, 1) \), \( (-\frac{5}{2}, \frac{4}{3}) \)

Step-by-Step Solution

Answer

The plotted points are \( (\frac{1}{2}, 1) \) and \( (-\frac{5}{2}, \frac{4}{3}) \) on the graph. The distance between the points is approximately 3.60555 after evaluating the distance formula. The midpoint of the line segment joining the points is approximately (-1, 1.16667) after evaluating the midpoint formula.

Step 1: Plotting the points

First plot the points \( (\frac{1}{2}, 1) \) and \( (-\frac{5}{2}, \frac{4}{3}) \) on a graph. Place the points on the correct x and y values.

Step 2: Find the distance between the points

Find the distance between the points using the distance formula for a plane which is given by \( D = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \). Substituting the given points into the formula, the distance becomes \( D = \sqrt{ [(-\frac{5}{2}-\frac{1}{2})^2 + (\frac{4}{3}-1)^2]} \). Simplifying the above expression will give the distance.

Step 3: Find the midpoint

The midpoint of the line segment joining the points can be found using the midpoint formula \( M = ( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} ) \). Substituting the given points into the formula, the midpoint becomes \( M = ( \frac{\frac{1}{2} - \frac{5}{2}}{2}, \frac{1 + \frac{4}{3}}{2} ) \). Simplifying the above expression will give the midpoint.

Key Concepts

Plotting Points

When beginning with the geometric visualization of mathematical problems, plotting points is the foundational skill. This involves placing points on the coordinate plane according to their x (horizontal) and y (vertical) values.
To accurately plot a point, first locate the x-value on the horizontal axis and then the y-value on the vertical axis. The point where these two values meet is the location of your point. For example, to plot \( (\frac{1}{2}, 1) \), you would move half a unit to the right of the origin along the x-axis and then move up by 1 unit along the y-axis. Similarly, plotting \( (-\frac{5}{2}, \frac{4}{3}) \) means moving 2.5 units to the left and approximately 1.33 units up from the origin.

Distance Formula

The distance formula is a key concept when determining the length of the segment connecting two points on a coordinate plane. This formula, \( D = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \), derives from the Pythagorean theorem.

Take the x and y values of the first point and subtract them from those of the second point, square the results, and then sum the squares. Finally, take the square root of that sum to find the distance. It mirrors the process of finding the hypotenuse of a right-angled triangle, where the legs are the differences between the x and y values.

Midpoint Formula

The midpoint formula comes in handy when you need to find the exact center point of a line segment defined by two endpoints on the coordinate plane. This formula, \( M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}) \), simply finds the average of the x and y coordinates of the endpoints.

Finding the midpoint involves adding the x-values of the two points, dividing the result by 2 for the x-coordinate of the midpoint, and doing similarly with the y-values for the midpoint's y-coordinate. The result is a new point that is equidistant from both endpoints along the line segment.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface where each point is defined by an x-coordinate and a y-coordinate. It is divided into four quadrants by a horizontal axis (the x-axis) and a vertical axis (the y-axis).

Understanding the coordinate plane is crucial as it serves as the setting for plotting points, graphing lines, and working with geometric shapes. Each quadrant has a specific sign for x and y coordinates that helps in identifying the precise location of points. Quadrant I has both positive x and y values, Quadrant II has a negative x and positive y, Quadrant III has both negative, and Quadrant IV has a positive x and negative y.