Find the midpoint of segment AB if A(-3, 8) and B (-7, -6).

The Correct Answer and Explanation is:

To find the midpoint of a line segment between two points ( A(x_1, y_1) ) and ( B(x_2, y_2) ), we use the midpoint formula:

[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
]

In this problem, the coordinates of point ( A ) are ( (-3, 8) ) and the coordinates of point ( B ) are ( (-7, -6) ).

Step 1: Plug in the values into the midpoint formula

Given:

( x_1 = -3 ), ( y_1 = 8 ) (coordinates of point ( A ))
( x_2 = -7 ), ( y_2 = -6 ) (coordinates of point ( B ))

The formula for the midpoint is:

[
M = \left( \frac{-3 + (-7)}{2}, \frac{8 + (-6)}{2} \right)
]

Step 2: Perform the calculations

For the ( x )-coordinate:

[
\frac{-3 + (-7)}{2} = \frac{-10}{2} = -5
]

For the ( y )-coordinate:

[
\frac{8 + (-6)}{2} = \frac{2}{2} = 1
]

Step 3: Conclusion

Thus, the midpoint ( M ) of segment ( AB ) is:

[
M = (-5, 1)
]

Explanation

The midpoint formula calculates the average of the ( x )-coordinates and ( y )-coordinates of two points. The midpoint represents the point that is exactly halfway between the two endpoints of the segment, which makes it a valuable concept in geometry and coordinate plane analysis. The midpoint is important because it provides the central location between two points, making it useful for various applications such as dividing a segment into two equal parts or determining the center of a shape.

In this case, the midpoint ( (-5, 1) ) represents a point on the coordinate plane that lies halfway between points ( A(-3, 8) ) and ( B(-7, -6) ). By averaging the ( x )-coordinates and the ( y )-coordinates separately, we ensure that the midpoint is equidistant from both points. This method of finding midpoints is fundamental for understanding the geometric properties of segments and is widely used in problems involving distances, symmetry, and coordinate transformations.