Problem 8 - Algebra and Trigonometry

Question

\(7-14\) 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. $$ (-2,5),(10,0) $$

Step-by-Step Solution

Answer

The distance is 13, and the midpoint is (4, 2.5).

Step 1: Plot the Points

Start by plotting the two given points on the coordinate plane. The first point is \((-2, 5)\) and the second point is \((10, 0)\). To plot these points, move left 2 units and up 5 units for the first point, and right 10 units with no upward movement for the second point.

Step 2: Use the Distance Formula

To find the distance between the two points, use the distance formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substituting the given points \((-2, 5)\) and \((10, 0)\), we have:\[d = \sqrt{(10 - (-2))^2 + (0 - 5)^2}\]\[d = \sqrt{(10 + 2)^2 + (-5)^2}\]\[d = \sqrt{12^2 + 25}\]\[d = \sqrt{144 + 25}\]\[d = \sqrt{169}\]\[d = 13\]

Step 3: Use the Midpoint Formula

Find the midpoint using the midpoint formula:\[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]Substitute \((-2, 5)\) and \((10, 0)\) into the formula:\[M = \left( \frac{-2 + 10}{2}, \frac{5 + 0}{2} \right)\]\[M = \left( \frac{8}{2}, \frac{5}{2} \right)\]\[M = (4, 2.5)\]Thus, the midpoint is \((4, 2.5)\).

Key Concepts

Distance Formula

The distance formula is a handy tool for finding the distance between two points on a coordinate plane. This formula is derived from the Pythagorean theorem and can be used to calculate the straight-line distance, often called as "Euclidean distance."

To compute the distance, you first need the coordinates of both points. Consider the points are

• First point: \( (x_1, y_1) \)
• Second point: \( (x_2, y_2) \)

The distance formula is then written as: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] Plug the coordinates into the formula to calculate the distance.

Let's say the coordinates given are \((-2, 5)\) and \((10, 0)\). Use the formula:

• \(d = \sqrt{(10 - (-2))^2 + (0 - 5)^2} \)
• \(d = \sqrt{12^2 + (-5)^2} \)
• \(d = \sqrt{144 + 25} \)
• \(d = \sqrt{169} = 13 \)

Thus, the distance between the two points is 13 units. This method provides a clear way to determine how far apart the points are on a grid.

Midpoint Formula

The midpoint formula helps to find the exact middle point between two given points on the coordinate plane. The midpoint formula gives you the average position of the two endpoints.

To find the midpoint, use the coordinates of the endpoints. Assume these points are:

• First point: \( (x_1, y_1) \)
• Second point: \( (x_2, y_2) \)

The midpoint formula is expressed as: \[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\] You substitute your coordinates in to find the midpoint.

For points \((-2, 5)\) and \((10, 0)\), calculation proceeds as follows:

• \(M = \left( \frac{-2 + 10}{2}, \frac{5 + 0}{2} \right)\)
• \(M = \left( \frac{8}{2}, \frac{5}{2} \right)\)
• \(M = (4, 2.5) \)

Thus, the midpoint is \((4, 2.5)\), dividing the segment equally into two halves. This point represents the middle of the segment joining the two coordinates.

Plotting Points on a Coordinate Plane

Plotting points on a coordinate plane is the fundamental step in understanding how to visually represent data. The coordinate plane is a two-dimensional surface formed by the intersection of a horizontal line known as the x-axis and a vertical line called the y-axis. These axes divide the plane into four quadrants.

To plot a point, you'll need an ordered pair like this:

• Point: \( (x, y) \)

Here,

• \(x\) is the horizontal position and \(y\) is the vertical position.

For example, to plot the point \((-2, 5)\), start at the origin \((0, 0)\), move 2 units left on the x-axis, and then move up 5 units on the y-axis.
For \((10, 0)\), start again at the origin, move to the right 10 units on the x-axis, as there's no movement on the y-axis.

Plotting these points accurately is necessary for applying further tools like the distance and midpoint formulas. It lays the groundwork for solving problems in coordinate geometry.