Problem 10 - College Algebra and Calculus: An Applied Approach

Question

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \((37.5,-12.3),(-6.2,5.9)\)

Step-by-Step Solution

Answer

The distance between the points \((37.5, -12.3)\) and \((-6.2, 5.9)\) when calculated turns out to be approximately 46.27 units and the midpoint of the line segment joining of these points is approximately \((15.65, -3.2)\).

Step 1: Plotting the points

Create a coordinate plane and plot the points \((37.5,-12.3)\) and \((-6.2,5.9)\). Both points are represented as ordered pairs of the form (X,Y). Mark these points on the plane.

Step 2: Calculation of the Distance Between Points

Using the distance formula: \(D = \sqrt{{(x_2 - x_1)}^2 + {(y_2 - y_1)}^2}\), where \(x_1, y_1\) are the coordinates of the first point and \(x_2, y_2\) are the coordinates of the second point. Substitute the values in the formula and solve it: \(D = \sqrt{{(-6.2 - 37.5)}^2 + {(5.9 + 12.3)}^2}\). Calculate the values inside the brackets, square them, add these values, and finally take a square root of the result.

Step 3: Calculation of the Midpoint

The midpoint M of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \(M = \left(\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2}\right)\). Substitute the given values into the formula: \(M = \left(\frac{{37.5 - 6.2}}{2}, \frac{{-12.3 + 5.9}}{2}\right)\). Calculate the values inside brackets and divide them by two. This gives the coordinates of the midpoint.

Key Concepts

Plotting Points

Understanding how to plot points is an essential skill in coordinate geometry. To start plotting a point, imagine a two-dimensional grid with a horizontal line (the x-axis) and a vertical line (the y-axis). These axes intersect at a point called the origin, which has coordinates (0,0).

Each point on the plane is defined by an ordered pair (x, y), where 'x' is the horizontal value and 'y' is the vertical value. To plot the point \( (37.5,-12.3) \) for instance, start at the origin. Move 37.5 units to the right, as positive values indicate a rightward movement along the x-axis. Then move 12.3 units down, as negative values indicate a downward movement on the y-axis. Mark this position with a dot. Similarly, to plot \( (-6.2,5.9) \) move 6.2 units to the left and 5.9 units up from the origin, and place a second dot.

Use a straightedge to draw the x and y axes, ensuring they are perpendicular. Carefully measure along the axes to get the precise location for your points. Remember to label your points, which helps identify them and avoids confusion when you're dealing with multiple points on the same graph.

Distance Formula

The distance formula is a powerful tool derived from the Pythagorean theorem that calculates the exact length of the line segment connecting two points on the coordinate plane. It reflects the distance 'as the crow flies,' meaning directly from one point to another, regardless of the axes.

The formula is given by \[ D = \sqrt{{(x_2 - x_1)}^2 + {(y_2 - y_1)}^2} \], where \( x_1, y_1 \) and \( x_2, y_2 \) are the coordinates of two distinct points. To find the distance between \( (37.5,-12.3) \) and \( (-6.2,5.9) \) using the formula, subtract the x-coordinates of the first point from the second, and the y-coordinates of the first point from the second. Then, square these results, add them together, and take the square root of the sum defining the straight-line distance between the two points.

• For the x-coordinates: \((-6.2 - 37.5)^2 = (-43.7)^2 = 1910.69\).
• For the y-coordinates: \((5.9 + 12.3)^2 = (18.2)^2 = 331.24\).

Add these squared differences together and take their square root: \[ D = \sqrt{1910.69 + 331.24} = \sqrt{2241.93} \], yielding the distance. Ensure you carefully follow these steps, as errors in sign or in the order of operations can lead to incorrect results.

Midpoint Formula

The midpoint formula is just as important as the distance formula in coordinate geometry and is used for finding the exact center point between two locations on the coordinate plane.

This formula is expressed as \( M = \left(\frac{{x_1 + x_2}}{2}, \frac{{y_1 + y_2}}{2}\right) \). It determines the coordinates of the midpoint by averaging the x-coordinates and the y-coordinates of the endpoints respectively. For the points \( (37.5,-12.3) \) and \( (-6.2,5.9) \) merely add their x-coordinates and divide by two, then do the same with the y-coordinates, resulting in the midpoint's coordinates.

For example:

• Average of the x-coordinates: \(\frac{{37.5 - 6.2}}{2} = 15.65\).
• Average of the y-coordinates: \(\frac{{-12.3 + 5.9}}{2} = -3.2\).

Thus, the midpoint \( M \) has the coordinates \( (15.65, -3.2) \). When plotting this on a coordinate plane, this point would lie exactly halfway between the two given points. Remember, the midpoint can be considered the balance point of the line segment joining two points and is particularly useful in geometric constructions and proofs.