what is the midpoint of the segment shown below (3,7) (2,-1)
The Correct Answer and Explanation is:
To find the midpoint of the segment with endpoints (3,7)(3, 7)(3,7) and (2,−1)(2, -1)(2,−1), we use the midpoint formula: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)
Plugging in the coordinates:x1=3,y1=7,×2=2,y2=−1x_1 = 3,\quad y_1 = 7,\quad x_2 = 2,\quad y_2 = -1×1=3,y1=7,×2=2,y2=−1Midpoint=(3+22,7+(−1)2)=(52,62)=(2.5,3)\text{Midpoint} = \left( \frac{3 + 2}{2}, \frac{7 + (-1)}{2} \right) = \left( \frac{5}{2}, \frac{6}{2} \right) = \left( 2.5, 3 \right)Midpoint=(23+2,27+(−1))=(25,26)=(2.5,3)
Answer:
The midpoint is (2.5, 3).
Explanation
The concept of a midpoint is fundamental in coordinate geometry. A midpoint is the exact center between two points on a line segment. In a two-dimensional plane, the midpoint is calculated by averaging the x-coordinates and the y-coordinates of the two endpoints separately. This gives you a new point that lies halfway between them.
In this case, the endpoints are (3, 7) and (2, -1). To find the x-coordinate of the midpoint, you take the x-values of both points — 3 and 2 — and find their average. This is done by adding them together to get 5 and then dividing by 2. That gives 2.5. Similarly, for the y-coordinate, you take 7 and -1, add them to get 6, and then divide by 2. That gives 3. So, the midpoint of the segment is (2.5, 3).
This point lies directly in the middle of the segment, meaning it is equidistant from both endpoints. You could verify this by calculating the distance from each endpoint to the midpoint using the distance formula. If done correctly, both distances will be equal.
Understanding how to find midpoints is useful in many areas of mathematics and science. It is often used in geometry problems, computer graphics, and physics. For example, if you want to split a line into equal parts, the midpoint gives you a perfect dividing point. It also plays a role in finding centers of geometric shapes and in constructing geometric figures accurately.
The midpoint formula can be extended to three dimensions as well by including the z-coordinate and averaging it the same way. In this two-dimensional case, though, you only need the x and y coordinates to determine the exact center between two points.
