Question

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points. \(\left(\frac{1}{2}, 1\right),\left(-\frac{5}{2}, \frac{4}{3}\right)\)

Short answer

The distance between the points is \(1.5\) and the midpoint of the line segment joining the points is \((-1, \frac{7}{6})\).

Step-by-step solution

Plot the Points

First, the points \(\left(\frac{1}{2}, 1\right)\) and \(\left(-\frac{5}{2}, \frac{4}{3}\right)\) need to be plotted on a graph. This first step gives a visual representation of where the points are relative to one another and allows to better understand the subsequent steps.

Find the Distance Between the Points

Next, use the distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) to find the distance between the two points. Here, \(x_1 = \frac{1}{2}\), \(y_1 = 1\), \(x_2 = -\frac{5}{2}\), \(y_2 = \frac{4}{3}\). Substituting these values in the formula, the result will be \(d = \sqrt{(\frac{-5}{2} - \frac{1}{2})^2 + (\frac{4}{3} - 1)^2}\). After simplifying, one can calculate that \(d = \sqrt{\frac{81}{36}} = \frac{9}{6} = 1.5\).

Find the Midpoint of the Line Segment Joining the Points

Now use the midpoint formula \((M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\) to find the midpoint of the line joining the two points. Here, \(x_1 = \frac{1}{2}\), \(y_1 = 1\), \(x_2 = -\frac{5}{2}\), \(y_2 = \frac{4}{3}\). Substituting these values in the formula will give \(M = \left(\frac{\frac{1}{2} + \left(-\frac{5}{2}\right)}{2}, \frac{1 + \frac{4}{3}}{2}\right)\). After simplifying, the midpoint \(M = \left(-1, \frac{7}{6}\right)\).