Answer :
To find the location of the other stop sign given the coordinates of the fire hydrant and one stop sign, we can use the concept of the midpoint. The formula for the midpoint \((M_x, M_y)\) of a segment connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
[tex]\[ M_x = \frac{x_1 + x_2}{2} \quad \text{and} \quad M_y = \frac{y_1 + y_2}{2} \][/tex]
In this problem, we know the coordinates of the fire hydrant and one stop sign:
- Fire hydrant: \((12, 7)\)
- First stop sign: \((3, 11)\)
Given that the fire hydrant is the midpoint, we can set up the midpoint formula to solve for the coordinates of the other stop sign \((x_2, y_2)\):
[tex]\[ (12, 7) = \left(\frac{3 + x_2}{2}, \frac{11 + y_2}{2}\right) \][/tex]
We can solve these equations separately to find \(x_2\) and \(y_2\).
First, solve for \(x_2\):
[tex]\[ 12 = \frac{3 + x_2}{2} \][/tex]
Multiply both sides by 2:
[tex]\[ 24 = 3 + x_2 \][/tex]
Subtract 3 from both sides:
[tex]\[ x_2 = 21 \][/tex]
Next, solve for \(y_2\):
[tex]\[ 7 = \frac{11 + y_2}{2} \][/tex]
Multiply both sides by 2:
[tex]\[ 14 = 11 + y_2 \][/tex]
Subtract 11 from both sides:
[tex]\[ y_2 = 3 \][/tex]
Therefore, the coordinates of the other stop sign are \((21, 3)\).
Now, we check the given multiple-choice options to find the correct answer:
A. \(\left(\frac{9}{2}, 2\right)\)
B. \(\left(\frac{15}{2}, 9\right)\)
C. \((21, 3)\)
D. \((-6, 15)\)
The correct option is:
C. [tex]\((21, 3)\)[/tex]
[tex]\[ M_x = \frac{x_1 + x_2}{2} \quad \text{and} \quad M_y = \frac{y_1 + y_2}{2} \][/tex]
In this problem, we know the coordinates of the fire hydrant and one stop sign:
- Fire hydrant: \((12, 7)\)
- First stop sign: \((3, 11)\)
Given that the fire hydrant is the midpoint, we can set up the midpoint formula to solve for the coordinates of the other stop sign \((x_2, y_2)\):
[tex]\[ (12, 7) = \left(\frac{3 + x_2}{2}, \frac{11 + y_2}{2}\right) \][/tex]
We can solve these equations separately to find \(x_2\) and \(y_2\).
First, solve for \(x_2\):
[tex]\[ 12 = \frac{3 + x_2}{2} \][/tex]
Multiply both sides by 2:
[tex]\[ 24 = 3 + x_2 \][/tex]
Subtract 3 from both sides:
[tex]\[ x_2 = 21 \][/tex]
Next, solve for \(y_2\):
[tex]\[ 7 = \frac{11 + y_2}{2} \][/tex]
Multiply both sides by 2:
[tex]\[ 14 = 11 + y_2 \][/tex]
Subtract 11 from both sides:
[tex]\[ y_2 = 3 \][/tex]
Therefore, the coordinates of the other stop sign are \((21, 3)\).
Now, we check the given multiple-choice options to find the correct answer:
A. \(\left(\frac{9}{2}, 2\right)\)
B. \(\left(\frac{15}{2}, 9\right)\)
C. \((21, 3)\)
D. \((-6, 15)\)
The correct option is:
C. [tex]\((21, 3)\)[/tex]
