‘Find the midpoint of the segment with the following endpoints (1,-10) and (8,-4)’
The Correct Answer and Explanation is:
To find the midpoint of a line segment, you can use the midpoint formula, which is:Midpoint=(x1+x22,y1+y22)\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)Midpoint=(2×1+x2,2y1+y2)
Where (x1,y1)(x_1, y_1)(x1,y1) and (x2,y2)(x_2, y_2)(x2,y2) are the coordinates of the two endpoints.
Given endpoints:
- (x1,y1)=(1,−10)(x_1, y_1) = (1, -10)(x1,y1)=(1,−10)
- (x2,y2)=(8,−4)(x_2, y_2) = (8, -4)(x2,y2)=(8,−4)
Now, applying the midpoint formula:
- For the x-coordinate of the midpoint:
x1+x22=1+82=92=4.5\frac{x_1 + x_2}{2} = \frac{1 + 8}{2} = \frac{9}{2} = 4.52×1+x2=21+8=29=4.5
- For the y-coordinate of the midpoint:
y1+y22=−10+(−4)2=−142=−7\frac{y_1 + y_2}{2} = \frac{-10 + (-4)}{2} = \frac{-14}{2} = -72y1+y2=2−10+(−4)=2−14=−7
Therefore, the midpoint of the segment is:(4.5,−7)\boxed{(4.5, -7)}(4.5,−7)
Explanation:
The midpoint of a segment is essentially the “average” of the x-coordinates and y-coordinates of the two endpoints. By finding the mean of the x-values and the mean of the y-values, you get the exact point that divides the segment into two equal lengths. This point is equidistant from both endpoints.
This method works for any two points in a two-dimensional coordinate system. It is often useful in geometry and can help you find the center of a shape or the balance point of a line.
