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SAT - Math - Coordinate geometry

3 points lie on an xy plane

3 points lie on an xy plane, with each point representing where a particular person lives. Amy lives at point (2,3); Brian lives at point (0,7); and Claire lives at point (-2,3). They all wish to meet up with each other for lunch, but at a location that is equidistant from each of their respective houses. At what point do they meet?
a. (0,4.5)
b. (0,13/3)
c. (0,3)
d. (1,5)
e. (-1,5)

 

The point equidistant from 3 points in a coordinate plane would be the circumcenter of the triangle formed by the three points.

Circumcenter is the intersection point of the perpendicular bisectors of the sides of the triangle.

Equation perpendicular bisector of (2,3) and (0,7)

(x-2)2 + (y-3)2 = (x-0)2 + (y-7)2

simplyfing and canceling second order terms

4 - 4x + 9 - 6y = 49 - 14y

4x - 8y +36 = 0

Similarly Equation of perpendicular bisector of  (2,3) and (-2,3)

(x-2)2 + (y-3)2 = (x+2)2 + (y-3)2

(x-2)2 = (x+2)2

only solution x = 0

simultaneously solving both equations

put x = 0 in first equation

8y = 36

y = 9/2

required point: (0,4.5)