Problem 62 - College Algebra

Question

In your own words, describe how to find the midpoint of a line segment if its endpoints are known.

Step-by-Step Solution

Answer

The midpoint of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( M = \left( \frac{{x_1+x_2}}{2}, \frac{{y_1+y_2}}{2} \right) \).

Step 1: Understand the Concept

The midpoint is the point that divides a line segment into two equal parts. When the coordinates of the endpoints in a two-dimensional space are known, the midpoint can be calculated using the formula: \( M = \left( \frac{{x_1+x_2}}{2}, \frac{{y_1+y_2}}{2} \right) \). Where \( M \) is the midpoint and \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the endpoints.

Step 2: Apply the Midpoint Formula

Let's suppose the coordinates of the endpoints are \( (x_1, y_1) \) and \( (x_2, y_2) \). Apply the midpoint formula \( M = \left( \frac{{x_1+x_2}}{2}, \frac{{y_1+y_2}}{2} \right) \). Calculate the sum of the x-coordinates (\( x_1 + x_2 \)) and the y-coordinates (\( y_1 + y_2 \)), then divide each by 2 to find the x-coordinate and y-coordinate of the midpoint.

Step 3: Finalize the Answer

The calculated x and y coordinates form the coordinates of the midpoint \( M \). So, the midpoint is \( M = \left( \frac{{x_1+x_2}}{2}, \frac{{y_1+y_2}}{2} \right) \).