To find the coordinates of the other end of the fence, we can use the midpoint formula. The midpoint [tex]\((M)\)[/tex] of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
In this problem, we are given:
- A starting point: [tex]\((8, 5)\)[/tex]
- A midpoint: [tex]\((3.5, -1)\)[/tex]
Let's call the coordinates of the other end of the fence [tex]\((x_2, y_2)\)[/tex].
According to the midpoint formula, we can set up two equations based on the given information:
1. [tex]\[ \frac{8 + x_2}{2} = 3.5 \][/tex]
2. [tex]\[ \frac{5 + y_2}{2} = -1 \][/tex]
We solve each equation for [tex]\(x_2\)[/tex] and [tex]\(y_2\)[/tex]:
For [tex]\(x_2\)[/tex]:
[tex]\[ \frac{8 + x_2}{2} = 3.5 \][/tex]
First, multiply both sides by 2 to clear the fraction:
[tex]\[ 8 + x_2 = 7 \][/tex]
Next, solve for [tex]\(x_2\)[/tex] by subtracting 8 from both sides:
[tex]\[ x_2 = 7 - 8 \][/tex]
[tex]\[ x_2 = -1 \][/tex]
For [tex]\(y_2\)[/tex]:
[tex]\[ \frac{5 + y_2}{2} = -1 \][/tex]
First, multiply both sides by 2 to clear the fraction:
[tex]\[ 5 + y_2 = -2 \][/tex]
Next, solve for [tex]\(y_2\)[/tex] by subtracting 5 from both sides:
[tex]\[ y_2 = -2 - 5 \][/tex]
[tex]\[ y_2 = -7 \][/tex]
Thus, the coordinates of the other end of the fence are:
[tex]\[ (x_2, y_2) = (-1, -7) \][/tex]
Among the given choices:
- [tex]\((2, -9)\)[/tex]
- [tex]\((-15, -3)\)[/tex]
- [tex]\((-1, -7)\)[/tex]
- [tex]\((12, 6)\)[/tex]
The correct answer is [tex]\((-1, -7)\)[/tex].
