A segment has a midpoint at (2, -7) and an endpoint at (8, -5). What are the coordinates of the other endpoint?
We are given a line segment with a known midpoint \(M = (2, -7)\) and one endpoint \(A = (8, -5)\). We need to find the coordinates of the other endpoint, let's call it \(B = (x, y)\).
The midpoint formula for a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]
In our case, \((x_1, y_1) = (8, -5)\) and \((x_2, y_2) = (x, y)\), and the midpoint is \((2, -7)\).
Using the midpoint formula, we set up two separate equations for the x-coordinate and the y-coordinate:
1) \(\frac{8 + x}{2} = 2\)
2) \(\frac{-5 + y}{2} = -7\)
Now, solve for \(x\):
Multiply both sides of the first equation by 2:
\(8 + x = 4\)
Subtract 8 from both sides:
\(x = 4 - 8 = -4\)
Now, solve for \(y\):
Multiply both sides of the second equation by 2:
\(-5 + y = -14\)
Add 5 to both sides:
\(y = -14 + 5 = -9\)
The coordinates of the other endpoint \(B\) are \((x, y) = (-4, -9)\).
To verify, check the midpoint of the segment with endpoints \((8, -5)\) and \((-4, -9)\):
Midpoint = \(\left( \frac{8 + (-4)}{2}, \frac{-5 + (-9)}{2} \right) = \left( \frac{4}{2}, \frac{-14}{2} \right) = (2, -7)\).
This matches the given midpoint.
The final answer is \(\boxed{(-4, -9)}\).
Imagine you are setting up a game board with a direct path between two objects. If you know where the center of the path is and where one object is located, you can easily calculate the position of the second object using the midpoint formula. For instance, if you have a path center at \((2, -7)\) and one object at \((8, -5)\), calculating the other end position as \((-4, -9)\) helps ensure your objects are placed symmetrically and correctly across the board.