Question

A new firehouse is being built equidistant from three other fire stations. Positioned on a grid, the current fire stations would be located at (3, 7), (−1, −1), and (−4, 8). What are the coordinates of the location where the new firehouse should be built? (PLEASE INCLUDE AN EXPLANATION)

Answer 1

`(-(3)/(22),(185)/(22) )`

Explanation:

Step 1: Find out what the question is asking

Because we want the point to be equidistant to all the vertices of the triangle formed we want the circumcenter of the triangle as it is equidistant from all of the vertices, a circumcenter is made from the perpendicular bisectors of the triangle

Step 2: Give the vertices variables

A = (-4,8)

B = (-1,-1)

C = (3,7)

Step 2: Find the Circumcenter

Find the mid point of side AB

MAB

`=((-4+(-1))/(2) ,(8+(-1))/(2) )\\=((3)/(2),(7)/(2) )`

Find the slope of side AB

mAB

`=(y2-y1)/(x2-x1) \\=(8-(-1))/(-4-(-1)) \\=(9)/(-3)\\ =-3`

Solve for 'b'

`((7)/(2))=(-3)((3)/(2) ) +b\\(7)/(2)=-(9)/(2)+b\\(7)/(2)+ (9)/(2)=b\\b = 8`

Therefore the equation for line AB is:

y = -3x + 8

Find the midpoint of side AC

MAC

`=((-4+(3))/(2) ,(8+(7))/(2) )\\=(-(1)/(2),(15)/(2) )`

Find slope of AC

mAC

`=(y2-y1)/(x2-x1) \\=(8-7)/(-4-3) \\=(1)/(-7)\\ =-(1)/(7)`

Solve for 'b'

`y=mx+b\\(15)/(2) =-(1)/(7) (-(1)/(2) )+b\\(15)/(2) =-(1)/(14) +b\\(1)/(14)+(15)/(2) = b\\b=(53)/(7)`

Therefore the equation for line AC is:

y =
`-(1)/(7)`x+
`(53)/(7)`

Step 3: Find the P.O.I of the 2 lines

Set both of the equation equal to each other

`3x + 8=-(1)/(7) x+(53)/(7) \\3x +(1)/(7) x=(53)/(7)-8\\(22)/(7) x=-(3)/(7) \\x=-(3)/(22)`

We now know 'x' and we substitute x into any equation to find 'y'

`y = -3(-(3)/(22) ) + 8\\y = (9)/(22) + 8\\y = (185)/(22)`

Therefore the new firehouse should be built at
`(-(3)/(22),(185)/(22) )`