To find the coordinates of the other end of Grady's father's fence, we need to use the midpoint formula and solve for the unknown endpoint.
The midpoint formula is given by:
[tex]\[
\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] is the start point, and [tex]\((x_2, y_2)\)[/tex] is the endpoint of the line segment.
Given:
- Start point [tex]\((x_1, y_1) = (8, 5)\)[/tex]
- Midpoint [tex]\((\text{mid}_x, \text{mid}_y) = (3.5, -1)\)[/tex]
To find the coordinates of the other end point [tex]\((x_2, y_2)\)[/tex], we will use the midpoint formula and solve for [tex]\(x_2\)[/tex] and [tex]\(y_2\)[/tex].
Step 1: Solve for [tex]\(x_2\)[/tex]
Using the midpoint formula for the x-coordinates:
[tex]\[
3.5 = \frac{8 + x_2}{2}
\][/tex]
Multiply both sides by 2 to isolate [tex]\(x_2\)[/tex]:
[tex]\[
7 = 8 + x_2
\][/tex]
Subtract 8 from both sides:
[tex]\[
x_2 = 7 - 8
\][/tex]
[tex]\[
x_2 = -1
\][/tex]
Step 2: Solve for [tex]\(y_2\)[/tex]
Using the midpoint formula for the y-coordinates:
[tex]\[
-1 = \frac{5 + y_2}{2}
\][/tex]
Multiply both sides by 2 to isolate [tex]\(y_2\)[/tex]:
[tex]\[
-2 = 5 + y_2
\][/tex]
Subtract 5 from both sides:
[tex]\[
y_2 = -2 - 5
\][/tex]
[tex]\[
y_2 = -7
\][/tex]
Therefore, the coordinates of the other end of the fence are [tex]\((-1, -7)\)[/tex].
The correct answer is:
[tex]\[ \boxed{(-1, -7)} \][/tex]
