Determine the coordinates for fire station

Question

On a map, three towns are located at the coordinates A(5,1), B(-2,0) and C(4,8). The township wants to build a new fire station that is the same distance from each town. Determine the coordinates where they should build the new fire station. (Hint: The circumcentre is the same distance from each vertex).

Answer 1

Let P(x,y) be the point that is equidistant from the 3 towns.

Using the 2D distance formula:

√((x-5)² + (y-1)²) = √((x+2)² + (y-0)²) = √((x-4)² + (y-8)²)

Squaring all 3 expressions:

(x-5)² + (y-1)² = (x+2)² + (y-0)² = (x-4)² + (y-8)²

The squared terms drop out, and collecting like terms we have

-10x -2y + 26 = 4x + 4 = -8x -16y + 80

Or, as 3 equations:

-10x -2y + 26 = 4x + 4 Eqn A

4x + 4 = -8x -16y + 80 Eqn B

-10x -2y + 26 = -8x -16y + 80 Eqn C

Collecting like terms:

14x + 2y = 22 A'

12x + 16y = 76 B'

-2x + 14y = 54 C'

Multiply A' by 7, and subtract C':

100x = 100

So

x = 1

Substituting x = 1 in Eqn A, we get y = 4

So the equidistant point is at (1,4).

To verify, plug (1,4) into the 3 distance expressions (the distance is 5).

Answer 2

LINE SEGMENT AB:

midpoint of AB is (1,5. 0.5)

slope of AB is 1/7

perpendicular bisector has slope -7 and eats (1.5,0.5)

perpendicular bisector has intercept B = y - mx = 0.5 - (-7)(1.5)

= 0.5 + 10.5

= 11

perpendicular bisector has equation y = -7x + 11

Line segment BC:

midpoint (1,4)

slope is 4/3

perpendicular bisector has slope -3/4

perpendicular bisector has intercept B=y-mx = 4 - (-3/4)(1)

= 4- 3/4

= 3 and 1/4 = 13/4

Perpendicular bisector of BC has equation y = (-3/4)x + 13/4

y = -7x + 11 vs y (-3/4)x + 13/4

-7x + 11 = (-3/4)x + 13/4

-28x + 44 = -3x + 13

44 = 25x + 13

31 = 25x

x = 31/25

-7(31/25) + 11 = =217/25 + 11 = -217/25 + 275/25 = 58/25

(-3/4)(31/25) + 13/4 = -93/100 + 325/100 = 232/100 = 116/50 = 58/25

(31/25, 58/25)