Question

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \((-7,3),(2,-9)\)

Short answer

The plot of points will have (-7,3) and (2,-9). The distance between the points is 15. The midpoint of the line segment joining them is \((-2.5, -3)\).

Step-by-step solution

Plotting the Points

First, plot the points (-7,3) and (2,-9) on the Cartesian plane. The point (-7,3) means that you move 7 units to the left and 3 units upwards on the graph. The point (2,-9) means that you move 2 units to the right and 9 units downwards.

Finding the Distance

Next, find the distance between these two points using the distance formula, which is \(d = \sqrt{(x2-x1)^2 + (y2-y1)^2}\). Substitute \(-7\) for \(x1\), \(3\) for \(y1\), \(2\) for \(x2\) and \(-9\) for \(y2\), to get \(d = \sqrt{(2-(-7))^2 + ((-9)-3)^2}\). Thus, \(d = \sqrt{9^2 + (-12)^2} = \sqrt{81 + 144} = \sqrt{225} = 15\).

Finding the Midpoint

Finally, determine the midpoint of the two points. The formula for finding the midpoint is \((x,y) = ((x1 + x2)/2, (y1 + y2)/2)\). So, the midpoint between (-7,3) and (2,-9) is \((-7 + 2)/2, (3-9)/2\) and hence \(-5/2, -6/2\) = \(-2.5, -3\).

Key concepts

Midpoint Formula

When you have two points on the Cartesian plane, you might want to find the middle point, which is called the midpoint. The midpoint is like the average position between these two points.

The formula for finding the midpoint

• If you have two points \((x_1, y_1)\) and \((x_2, y_2)\), the midpoint is calculated using:\((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\)

Apply the midpoint formula

• Take the two x-coordinates and find their average: \((x_1 + x_2)/2\)
• Do the same for the y-coordinates: \((y_1 + y_2)/2\)

For example, with points \((-7, 3)\) and \((2, -9)\):

• Average of x-values: \((-7 + 2)/2 = -2.5\)
• Average of y-values: \((3 - 9)/2 = -3\)

This means the midpoint is \((-2.5, -3)\). Understanding this helps in visualizing how far and in which direction the midpoint is located on the plane.

Cartesian Plane

The Cartesian plane is a two-dimensional surface where you can plot points, lines, and shapes. It is made up of two perpendicular lines called axes:

• The horizontal line is called the x-axis.
• The vertical line is called the y-axis.

Understanding the Cartesian plane

The intersection of these axes is called the origin, denoted as \((0,0)\), which is the starting point for plotting any other point.

• To plot a point like \((-7, 3)\), move 7 units left from the origin and 3 units up.
• For \((2, -9)\), move 2 units right and 9 units down.

Using the Cartesian plane allows you to visualize mathematical concepts such as distance and midpoints, making them more tangible and easier to understand.

Line Segment

A line segment is the shortest path connecting two points on the Cartesian plane. Unlike a line, it does not extend infinitely but has fixed endpoints.

Properties of a line segment

• Defined by two endpoints, such as \((-7, 3)\) and \((2, -9)\)
• Has a measurable length, which can be found using the distance formula:

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

For example, the length of the line segment between \((-7, 3)\) and \((2, -9)\) is \(15\).

A line segment can also be used to find the midpoint, which helps in dividing it into two equal parts. Understanding line segments is crucial for grasping more complex geometric concepts.