To determine the location of the second stop sign, given the positions of the fire hydrant and one of the stop signs, we can use the midpoint formula.
The midpoint formula states that the midpoint [tex]\( M \)[/tex] between two points [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]
From the problem, we have the coordinates of the fire hydrant (midpoint), [tex]\( M = (12,7) \)[/tex], and the coordinates of one stop sign, [tex]\( (3,11) \)[/tex]. Let's denote the unknown coordinates of the second stop sign as [tex]\( (x_2, y_2) \)[/tex].
Using the formula for the midpoint, we can set up the following equations for the [tex]\( x \)[/tex]-coordinate and [tex]\( y \)[/tex]-coordinate separately.
For the [tex]\( x \)[/tex]-coordinates:
[tex]\[ 12 = \frac{3 + x_2}{2} \][/tex]
Solving for [tex]\( x_2 \)[/tex]:
[tex]\[ 24 = 3 + x_2 \][/tex]
[tex]\[ x_2 = 21 \][/tex]
For the [tex]\( y \)[/tex]-coordinates:
[tex]\[ 7 = \frac{11 + y_2}{2} \][/tex]
Solving for [tex]\( y_2 \)[/tex]:
[tex]\[ 14 = 11 + y_2 \][/tex]
[tex]\[ y_2 = 3 \][/tex]
Thus, the coordinates of the second stop sign are [tex]\( (21, 3) \)[/tex].
Therefore, the correct answer is:
C. [tex]\( (21, 3) \)[/tex]
