Problem 15 - Precalculus Mathematics for Calculus

Question

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. $$(-3,-6),(4,18)$$

Step-by-Step Solution

Answer

Distance: 25 units; Midpoint: (0.5, 6).

Step 1: Plot the Points

First, we need to plot the points (-3, -6) and (4, 18) on a coordinate plane. The point (-3, -6) is located 3 units left and 6 units down from the origin, and the point (4, 18) is 4 units right and 18 units up from the origin.

Step 2: Calculate the Distance

To find the distance between two points (x_1, y_1) and (x_2, y_2), we use the distance formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substituting our points (-3, -6) and (4, 18) into the formula, we have:\[d = \sqrt{(4 - (-3))^2 + (18 - (-6))^2}\]\[d = \sqrt{(4 + 3)^2 + (18 + 6)^2}\]\[d = \sqrt{7^2 + 24^2}\]\[d = \sqrt{49 + 576}\]\[d = \sqrt{625}\]\[d = 25\]So, the distance between the points is 25 units.

Step 3: Calculate the Midpoint

To find the midpoint of the segment joining two points (x_1, y_1) and (x_2, y_2), we use the midpoint formula:\[m = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]Substituting our points (-3, -6) and (4, 18) into the formula, we get:\[m = \left(\frac{-3 + 4}{2}, \frac{-6 + 18}{2}\right)\]\[m = \left(\frac{1}{2}, \frac{12}{2}\right)\]\[m = \left(0.5, 6\right)\]Therefore, the midpoint of the segment that joins the points is (0.5, 6).

Key Concepts

Plotting Points

Plotting points on a coordinate plane is an essential skill in coordinate geometry. The coordinate plane is divided into four quadrants by the x-axis (horizontal) and y-axis (vertical). Each point is identified by a pair of coordinates (x, y), where 'x' shows the horizontal position, and 'y' shows the vertical position.
To plot a point:

• Identify the x-coordinate and move right (positive) or left (negative) from the origin.
• Identify the y-coordinate and move up (positive) or down (negative) from the origin.

For example, to plot the point (-3, -6), you move three units left and six units down from the origin (0,0). Similarly, for the point (4, 18), you move four units right and eighteen units up. Practice plotting these points on graph paper to better understand their positions.

Distance Formula

The distance formula is a mathematical equation used to calculate the distance between two points in a coordinate plane. It is derived from the Pythagorean theorem and is essential for finding precise measurements between points.
The formula is:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Here, \((x_1, y_1)\) and \((x_2, y_2)\) represent the coordinates of the two points. To apply the distance formula:

• Subtract the x-coordinates \((x_2 - x_1)\) and y-coordinates \((y_2 - y_1)\) of the two points.
• Square both differences.
• Add the squared differences together.
• Take the square root of the sum to get the distance.

Using this formula ensures accuracy when determining the distance, as shown in our example where the distance calculated is 25 units.

Midpoint Formula

The midpoint formula helps find the center point of a line segment connecting two points on the coordinate plane. It's like the average of the points' coordinates and is particularly useful in geometry for identifying central locations.
The formula is:\[m = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]Where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. Applying the midpoint formula involves:

• Adding the x-coordinates \((x_1 + x_2)\) and dividing by 2 to find the midpoint's x-coordinate.
• Adding the y-coordinates \((y_1 + y_2)\) and dividing by 2 to find the midpoint's y-coordinate.

In the exercise, using this method shows that the midpoint between points (-3, -6) and (4, 18) is (0.5, 6). This concept helps understand balance and symmetry on a graph.