Question

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \((-36,-18),(48,-72)\)

Short answer

The distance between points \((-36,-18)\) and \( (48,-72) \) is approximately 117.44 units. The midpoint of the line segment joining these points is given by the coordinates \((6,-45)\).

Step-by-step solution

Plot the Points

Plot the given points, \( (-36,-18) \) and \( (48,-72) \), on a graph. An understanding of how to mark points on a plane according to their coordinates is required here.

Calculate the Distance between Two Points

Use the Euclidean distance formula, which is \(\sqrt{{(x_2-x_1)^2+(y_2-y_1)^2}}\). Substitute the given points into the formula: \(\sqrt{{(48-(-36))^2+(-72-(-18))^2}}\). After the calculation, the distance between these points is found.

Calculate the Midpoint of the Line Segment

Use the midpoint formula, which is \((\frac{{x_1+x_2}}{2},\frac{{y_1+y_2}}{2})\). Substitute the given points into the formula: \((\frac{{-36+48}}{2},\frac{{-18+(-72)}}{2})\). After the calculation, the coordinates for the midpoint of the line segment are found.

Key concepts

Plotting Points

Understanding how to plot points on a coordinate plane is a fundamental skill in geometry. Coordinates are usually written in the form \( (x, y) \), where \(x\) represents the horizontal position and \(y\) the vertical position. To plot a point, you start at the origin \( (0, 0) \) and move \(x\) units along the horizontal axis, and \(y\) units along the vertical axis.

For example, to plot the point \( (-36, -18) \) you move 36 units to the left (since it's negative) from the origin along the x-axis and then 18 units down (also negative) along the y-axis. Similarly, you plot \( (48, -72) \) by moving 48 units to the right and 72 units down.

Distance Formula

The distance formula is used to calculate the straight-line distance between two points in Euclidean space (i.e., a flat surface). It is derived from the Pythagorean theorem and written as \(\sqrt{{(x_2-x_1)^2+(y_2-y_1)^2}}\).

To apply the distance formula, simply subtract the \(x\) and \(y\) values of the first point from the corresponding values of the second point, square the results, add them together, and take the square root of the sum. This formula tells you the length of the hypothetical line segment connecting the two points, giving you a precise measure of how far apart they are.

Midpoint Formula

The midpoint of a line segment is precisely in the middle of the two endpoints. It can easily be found using the midpoint formula, which is \((\frac{{x_1+x_2}}{2}, \frac{{y_1+y_2}}{2})\). This formula averages the \(x\)-coordinates and the \(y\)-coordinates of the two points, respectively, giving you the coordinates of the midpoint.

When you have two points, like our example with \( (-36, -18) \) and \( (48, -72) \), you add the \(x\)-coordinates together and divide by two for the \(x\)-coordinate of the midpoint, and do the same with the \(y\)-coordinates for the \(y\)-coordinate of the midpoint. This calculation provides the exact location halfway between the two points on the coordinate plane.

Euclidean Distance

Euclidean distance refers to the length of a straight line between two points in a Euclidean plane. It is the most common way of representing the distance between points in many fields, including mathematics and computer science. The Euclidean distance ensures the shortest path between two points, which is practical for various applications from geometry to optimization problems.

This concept is essential as it forms the basis for many other complex geometrical computations, such as in the fields of vector calculus or computer graphics, where distances between objects need to be found accurately and quickly.

Graphing Coordinates

Graphing coordinates involves placing points represented by pairs of numbers onto a two-dimensional grid. This grid is made up of two number lines: the x-axis, which runs horizontally, and the y-axis, which runs vertically. Their intersection represents the origin, or \( (0, 0) \).

When graphing, think of the coordinate plane as a map, with the x-axis indicating a location east or west of the origin, and the y-axis indicating a location north or south. The coordinates tell us how to navigate from the origin to reach the point. Students typically start by plotting simple points and gradually move on to more complex operations involving these points, such as finding distances and midpoints, or graphing lines and curves.