Problem 56 - Precalculus with Limits

Question

In Exercises 47-56, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \( (-16.8, 12.3) \), \( (5.6, 4.9) \)

Step-by-Step Solution

Answer

After plotting the points, the distance is calculated as approximately 23.03 units, and the midpoint is at approximately (-5.6, 8.6).

Step 1: Plotting the Points

Starting with the points given (-16.8, 12.3) and (5.6, 4.9), make a sketch of the 2D Cartesian plane and mark these points.

Step 2: Calculating the Distance

To calculate the distance between the points, use the distance formula: \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Substitute \(x_1 = -16.8, x_2 = 5.6, y_1 = 12.3, y_2 = 4.9\), to get the distance equal to \(\sqrt{(5.6 - (-16.8))^2 + (4.9 - 12.3)^2}\). Simplify this expression to calculate the distance.

Step 3: Calculating the Midpoint

The midpoint of a line segment can be found by averaging the x-coordinates and the y-coordinates of the endpoints. The formula is \(\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)\). Substitute \(x_1 = -16.8, x_2 = 5.6, y_1 = 12.3, y_2 = 4.9\), to find the midpoint: \(\left(\dfrac{-16.8+5.6}{2}, \dfrac{12.3+4.9}{2}\right)\). Simplify this expression to get the coordinates of the midpoint.

Key Concepts

Plotting Points

Understanding the basics of plotting points is essential in coordinate geometry, as it's the foundation upon which more complex concepts are built. To plot a point, imagine a flat surface split into four quadrants by a horizontal line (the x-axis) and a vertical line (the y-axis). These lines intersect at the origin, designated as (0,0).

Each point is given as an ordered pair \( (x, y) \), where \( x \) corresponds to a position along the horizontal axis, and \( y \) corresponds to a position along the vertical axis. To plot the point \( (-16.8, 12.3) \), start from the origin, move 16.8 units to the left (as the x-value is negative), then 12.3 units up (since the y-value is positive). Similarly, \( (5.6, 4.9)\) would be plotted by moving 5.6 units right and 4.9 units up from the origin.

Visualizing this process is vital: draw a Cartesian plane on paper, mark the axes, and then plot the points. Practice by plotting a variety of points to understand how the position of a point changes with different x and y values.

Distance Formula

The distance formula is a powerful tool used to find the length of the line segment connecting two points on a Cartesian plane. It's derived from the Pythagorean theorem and can be expressed as \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.

To apply this formula to our example with points \( (-16.8, 12.3) \) and \( (5.6, 4.9) \), identify \(x_1 = -16.8\), \(x_2 = 5.6\), \(y_1 = 12.3\), and \(y_2 = 4.9\). Replace these values in the formula to find the distance: \( \sqrt{(5.6 - (-16.8))^2 + (4.9 - 12.3)^2} \). Simplifying the expression provides the precise distance between the two points. Remember that the distance is always a positive value, representing the length of the straight line path between the points.

Midpoint Formula

Similar to the distance formula, the midpoint formula is just as crucial in coordinate geometry, particularly when you need to determine the exact center between two points. The formula \( \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \) calculates the average of the x-coordinates and the y-coordinates of the endpoints.

Taking the points \( (-16.8, 12.3)\) and \( (5.6, 4.9)\), we substitute the values into the midpoint formula: \( \left(\frac{-16.8+5.6}{2}, \frac{12.3+4.9}{2}\right) \). The midpoint then is the result of these calculations. This point is significant because it represents the exact halfway mark of the line segment connecting the two original points. Whether you're constructing bisectors, analyzing shapes, or merely finding the center, the midpoint formula is an essential aspect of coordinate geometry.