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Kelsey's neighborhood has a straight road with stop signs at both ends and a fire hydrant at its midpoint. On a map, the fire hydrant is located at [tex]\((12, 7)\)[/tex] and one of the stop signs is located at [tex]\((3, 11)\)[/tex]. Where on the map is the other stop sign located?

A. [tex]\(\left(\frac{9}{2}, 2\right)\)[/tex]

B. [tex]\(\left(\frac{15}{2}, 9\right)\)[/tex]

C. [tex]\((21, 3)\)[/tex]

D. [tex]\((-6, 15)\)[/tex]



Answer :

GinnyAnswer
To solve this problem, we need to determine the coordinates of the second stop sign using the given coordinates of the fire hydrant and the first stop sign. Given that the fire hydrant is at the midpoint between the two stop signs, we can use the midpoint formula, which states that the midpoint [tex]\((x_m, y_m)\)[/tex] of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated as follows:

[tex]\[ x_m = \frac{x_1 + x_2}{2} \][/tex]
[tex]\[ y_m = \frac{y_1 + y_2}{2} \][/tex]

Given:
- Fire hydrant coordinates: [tex]\((12, 7)\)[/tex]
- First stop sign coordinates: [tex]\((3, 11)\)[/tex]

We need to find the coordinates [tex]\((x_2, y_2)\)[/tex] of the second stop sign. Using the midpoint formula, we can set up the following equations:

[tex]\[ 12 = \frac{3 + x_2}{2} \][/tex]
[tex]\[ 7 = \frac{11 + y_2}{2} \][/tex]

Solving these equations step-by-step:

1. For the x-coordinate:
[tex]\[ 12 = \frac{3 + x_2}{2} \][/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[ 24 = 3 + x_2 \][/tex]
Subtract 3 from both sides:
[tex]\[ x_2 = 21 \][/tex]

2. For the y-coordinate:
[tex]\[ 7 = \frac{11 + y_2}{2} \][/tex]
Multiply both sides by 2 to clear the fraction:
[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 [tex]\((21, 3)\)[/tex].

The correct answer is:

C. [tex]\((21, 3)\)[/tex]

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