Problem 76 - College Algebra Essentials

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 whose endpoints are \((x_1, y_1)\) and \((x_2, y_2)\) is found by using the formulas \((x_1 + x_2) / 2\) and \((y_1 + y_2) / 2\) for the x and y coordinates respectively. So, the midpoint is \((x_m, y_m)\) where \(x_m = (x_1 + x_2) / 2\) and \(y_m = (y_1 + y_2) / 2\).

Step 1: Identify the coordinates of the end points

Suppose the coordinates of the two endpoints are \((x_1, y_1)\) and \((x_2, y_2)\) respectively. This means that one end of the line segment is at point \((x_1, y_1)\) and the other end is at point \((x_2, y_2)\).

Step 2: Calculate the x-coordinate of the midpoint

The x-coordinate of the midpoint can be found by averaging the x-coordinates of the endpoints. This is done by adding \(x_1\) and \(x_2\) together, and then dividing by 2. This gives the formula \((x_1 + x_2) / 2\). Calculate the result of this formula to get the x-coordinate of the midpoint.

Step 3: Calculate the y-coordinate of the midpoint

Similarly, the y-coordinate of the midpoint can be found by averaging the y-coordinates of the endpoints. This is done by adding \(y_1\) and \(y_2\) together, and then dividing by 2. This gives the formula \((y_1 + y_2) / 2\). Calculate the result of this formula to get the y-coordinate of the midpoint.

Step 4: Determine the coordinates of the midpoint

After getting the x and y coordinates of the midpoint, you can present the coordinates of the midpoint as a pair \((x_m, y_m)\), where \(x_m\) is the result from Step 2 and \(y_m\) is the result from Step 3. The pair \((x_m, y_m)\) constitutes the coordinates of the midpoint of the line segment.