To find the coordinates of the other end of the fence, we can use the midpoint formula. The midpoint formula states that if you have a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\), the coordinates of the midpoint \(M\) are:
[tex]\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
In this problem, we are given:
- The start point \((x_1, y_1) = (8, 5)\)
- The midpoint \((M_x, M_y) = (3.5, -1)\)
We need to find the coordinates of the other end of the fence \((x_2, y_2)\).
First, we use the x-coordinates to find \(x_2\):
[tex]\[
\frac{8 + x_2}{2} = 3.5
\][/tex]
Multiply both sides by 2 to eliminate the fraction:
[tex]\[
8 + x_2 = 7
\][/tex]
Then, solve for \(x_2\):
[tex]\[
x_2 = 7 - 8 = -1
\][/tex]
Next, we use the y-coordinates to find \(y_2\):
[tex]\[
\frac{5 + y_2}{2} = -1
\][/tex]
Multiply both sides by 2 to eliminate the fraction:
[tex]\[
5 + y_2 = -2
\][/tex]
Then, solve for \(y_2\):
[tex]\[
y_2 = -2 - 5 = -7
\][/tex]
Thus, the coordinates of the other end of the fence are \((-1, -7)\).
The correct answer is:
[tex]\[
\boxed{(-1, -7)}
\][/tex]
