Problem 18 - 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. $$(0,-6),(5,0)$$

Step-by-Step Solution

Answer

Distance is \(\sqrt{61}\), midpoint is \((2.5, -3)\).

Step 1: Plot the Points

Begin by plotting the points on a coordinate plane. We have the points \((0, -6)\) and \((5, 0)\). The first point \((0, -6)\) is located on the y-axis 6 units below the origin, and the second point \((5, 0)\) is found on the x-axis 5 units to the right of the origin.

Step 2: Calculate the Distance

Use the distance formula to find the distance between the two points. The distance formula is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Substitute the points \((x_1, y_1) = (0, -6)\) and \((x_2, y_2) = (5, 0)\) into the formula: \[ d = \sqrt{(5 - 0)^2 + (0 - (-6))^2} = \sqrt{5^2 + 6^2} = \sqrt{25 + 36} = \sqrt{61} \]

Step 3: Find the Midpoint

The formula for the midpoint of a segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]Substitute the given coordinates into the formula: \[ M = \left( \frac{0 + 5}{2}, \frac{-6 + 0}{2} \right) = \left( \frac{5}{2}, \frac{-6}{2} \right) = \left( 2.5, -3 \right) \]

Key Concepts

Distance Formula

The distance formula is an essential tool in coordinate geometry. It helps us find the distance between two points on a coordinate plane. This formula is derived from the Pythagorean theorem and is given by:

• \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
• Here, the coordinates of the first point are \((x_1, y_1)\) and the second point are \((x_2, y_2)\).

To use the distance formula:1. Subtract the x-coordinates: \((x_2 - x_1)\)2. Subtract the y-coordinates: \((y_2 - y_1)\)3. Square both differences.4. Add the squares.5. Take the square root of the sum.For example, for the points \((0, -6)\) and \((5, 0)\), substituting into the formula gives:

• \[ d = \sqrt{(5 - 0)^2 + (0 - (-6))^2} = \sqrt{25 + 36} = \sqrt{61} \]

This calculated distance \(\sqrt{61}\) is between the two points. This method ensures accurate measurement of the straight line joining any two points on the plane.

Midpoint Formula

The midpoint formula is used to find the point exactly halfway between two other points on a coordinate plane. This is very useful for finding the center point of a line segment. The formula for calculating the midpoint \(M\) 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.

To find the midpoint, simply:1. Add both x-coordinates and divide by 2.2. Add both y-coordinates and divide by 2.For the points given in our example \((0, -6)\) and \((5, 0)\):

• \[ M = \left( \frac{0 + 5}{2}, \frac{-6 + 0}{2} \right) = \left( 2.5, -3 \right) \]

So, the midpoint is \((2.5, -3)\). This helps in various geometric applications like bisecting lines and locating the center in symmetry problems.

Plotting Points on a Coordinate Plane

Plotting points on a coordinate plane is a fundamental skill in geometry. A coordinate plane consists of two perpendicular lines called axes: the x-axis (horizontal) and the y-axis (vertical). Where they intersect is the origin (0,0).To plot a point:1. Start from the origin.2. Move horizontally along the x-axis to the x-coordinate.3. Then, move vertically to the y-coordinate.Example:

• Consider the point \((0, -6)\). This lies on the y-axis, 6 units below the origin.
• For \((5, 0)\), it is on the x-axis, 5 units to the right of the origin.

Both points are critical in determining lines and segments between them. This process becomes the basis for many operations in coordinate geometry, such as creating graphs and helping visualize relationships between points.