Geometry Problem

Question

I have attempted this many times after help and I can't complete it. Help?

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

Hi Juan, Here are some suggestions. Try to determine the equations of the perpendicular bisectors of two sides of the triangle. That will require finding the midpoint of two of the sides and the slope of two of the sides. The perpendicular bisectors will have slopes that are negative reciprocals of the slopes of the corresponding sides.

 

Once you have those two equations, find the point of intersection. That is where the splash pad should be placed. To convince yourself find the distance between the determined point and two (or even all three) of the vertices. See the following for a picture:

 

https://www.desmos.com/calculator/yzcne8lqtc

 

Notice that the circle passes through all three vertices of the triangle. The center of the circle is the point of intersection of the two perpendicular bisectors. The radius of the circle is the distance from the center to one of the vertices.