What is the midpoint between (3,-1) and (8,-6)
The Correct Answer and Explanation is:
To find the midpoint between two points, you use the midpoint formula, which is:
[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
]
Where:
- ( (x_1, y_1) ) is the first point (3, -1),
- ( (x_2, y_2) ) is the second point (8, -6).
Step-by-Step Solution:
- Identify the coordinates of the two points:
- First point ( (x_1, y_1) = (3, -1) )
- Second point ( (x_2, y_2) = (8, -6) )
- Apply the midpoint formula:
- For the x-coordinate of the midpoint:
[
\frac{x_1 + x_2}{2} = \frac{3 + 8}{2} = \frac{11}{2} = 5.5
] - For the y-coordinate of the midpoint:
[
\frac{y_1 + y_2}{2} = \frac{-1 + (-6)}{2} = \frac{-7}{2} = -3.5
]
- Write the midpoint:
The midpoint is ( M = (5.5, -3.5) ).
Explanation:
The midpoint of a line segment is the point that is equidistant from both endpoints. It is found by averaging the x-coordinates and the y-coordinates of the two points. This is useful for determining the central point between two locations on a plane, which could represent anything from finding the center of a route between two cities to dividing a geometric figure in half. The midpoint formula ensures that the result is precisely halfway along the line segment connecting the two points.
In this case, the x-coordinates (3 and 8) average to 5.5, and the y-coordinates (-1 and -6) average to -3.5. Therefore, the midpoint between the points (3, -1) and (8, -6) is ( (5.5, -3.5) ).