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 (2,2) , (3, -5) , and (-5,-5). Find the coordinates of the new firehouse.

Answer 1

(-1, -2)

Explanation:

The point on the grid that is equidistant from the three given points is the center of a circle through those three points. That center is the point of intersection of the perpendicular bisectors of the sides of the triangle connecting the given points.

__

For points A(2, 2), B(3, -5), and C(-5, -5), we want to find the intersection of the perpendicular bisectors of AB, BC, and AC. It is convenient to choose pairs of points that make the computation easier. Points A and C have identical x- and y-values, so they lie on the line y=x. Points B and C have identical y-values, so lie on the horizontal line y=-5.

AC

As we observed, the line AC has equation y=x, so has a slope of 1. Its perpendicular will have a slope of -1. The midpoint of AC is ...

midpoint AC = (A +C)/2 = ((2, 2) +(-5, -5))/2 = (-3/2, -3/2)

In point-slope form, the equation of the perpendicular bisector of AC is ...

y -k = m(x -h) . . . . . . line with slope m through point (h, k)

y -(-3/2) = -1(x -(-3/2)) . . . . line with slope -1 through (-3/2, -3/2)

y = -x -3 . . . . . . . . . simplify to slope-intercept form

BC

The midpoint of BC is ...

midpoint BC = (B +C)/2 = ((3, -5) +(-5, -5))/2 = (-2, -10)/2 = (-1, -5)

The perpendicular to the horizontal line y = -5 will be a vertical line. It must pass through the midpoint we found, so its equation will be ...

x = -1

new firehouse

The circle center is the solution to the system ...

• y = -x -3
• x = -1

That solution will be ...

(x, y) = (-1, -(-1) -3)) = (-1, -2)

The new firehouse coordinates are (-1, -2).