A gardener wants the three rosebushes in her garden to be watered by a rotating water sprinkler. The gardener draws a diagram of the garden using a grid in which each unit represents 1 ft. The rosebushes are at (1, 3), (5, 11), and (11, 4). She wants to position the sprinkler at a point equidistant from each rosebush. Where should the gardener place the sprinkler? What equation describes the boundary of the circular region that the sprinkler will cover?

Question

A gardener wants the three rosebushes in her garden to be watered by a rotating water sprinkler. The gardener draws a diagram of the garden using a grid in which each unit represents 1 ft. The rosebushes are at (1, 3), (5, 11), and (11, 4). She wants to position the sprinkler at a point equidistant from each rosebush. Where should the gardener place the sprinkler? What equation describes the boundary of the circular region that the sprinkler will cover?

[It's okay if you'll only provide the formula needed. Thank you so much in advance! I'm really bad at math, so I'll appreciate if someone would explain it rather providing only the answers.]

Answer 1

presumably the sprinkler covers a circular area.

You can find the center of the circle either by

(a) solving (x-h)^2 + (y-k)^2 = r^2 using each of the three points
(b) find whee the perpendicular bisectors of the sides of the triangle, with vertices at those three points, intersect.

Both involve about the same amount of computation.

Or, you can go online and find the formula to get the circumcenter of a triangle, given its vertices.