Splash Pad geometry Problem

Question

Three straight paths in a park create a triangle with picnic shelters at their vertices. The picnic shelters are located at (-24, 16), (56, -16) and (-72, -32). Where should a "splash pad" be placed if it is to be the same distance from each of the picnic shelters.

Answer 1

Let A = (-24,16)

Let B = (56, -16)

Let C = (-72, -32)

 

Midpoint of AB = ((56-24)/2, (-16+16)/2)) =  (16,0)

 

Slope of segment AB:

m = (-16-16)/(56-(-24)) = -32/80 = -2/5

perpendicular slope is negative reciprocal = 5/2

 

equation for perpendicular bisector of AB is line through

(16,0) and slope 5/2

 

Using the point slope equation the equation for the perpendicular

bisector of AB is:

y - 0 = (5/2)(x-16)

y = (5/2)x - 40

2y = 5x - 80

5x - 2y = 80

 

Now find the midpoint of AC and BC

and their slopes in similar fashion.  Using the midpoints and

negative reciprocal slopes follow the same

procedure to determine the equations of

the perpendicular bisectors of AC and BC

 

This will give you three simultaneous equations

to solve.  The solution of this system of equations

will be the circumcenter of the system and the

answer to the problem

Answer 2

The point would be the circumcenter.

The circumcenter is the point at which the three perpendicular bisectors is the sides intersect.