A new firehouse is being built equidistant from three other fire stations. Positioned on a grid, the current fire stations woul
1 answer:
Answer:
(-1, -2)
Step-by-step 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.
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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.
<h3>AC</h3>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
<h3>BC</h3>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
<h3>new firehouse</h3>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).
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