Problem 27 - College Algebra

Question

Find the midpoint of each line segment with the given endpoints. $$(8,3 \sqrt{5}) \text { and }(-6,7 \sqrt{5})$$

Step-by-Step Solution

Answer

The midpoint of the line segment with endpoints (8, \(3\sqrt{5}\)) and (-6, \(7\sqrt{5}\)) is (1, \(5\sqrt{5}\)).

Step 1: Find average of x-coordinates

First, find the average of the x-coordinates of the endpoints. This is done by adding the x-coordinates together and dividing by 2. The x-coordinates of the endpoints are 8 and -6, so the average x-coordinate of the midpoint is \((8 + (-6)) / 2 = 1\).

Step 2: Find average of y-coordinates

Afterwards, find the average of the y-coordinates of the endpoints. This is also done by adding the y-coordinates together and dividing by 2. The y-coordinates of the endpoints are \(3\sqrt{5}\) and \(7\sqrt{5}\), so the average y-coordinate of the midpoint is \((3\sqrt{5} + 7\sqrt{5}) / 2 = 5\sqrt{5}\).

Step 3: State the midpoint

Finally, the midpoint is the point from the averages of the x-coordinates and y-coordinates. Thus the midpoint for the line segment with the given endpoints is (1, \(5\sqrt{5}\)).