Question

Big idea Coordinate Geometry\
You can use coordinates of the center of a circle and it radius to write an equation for a circle.

TASK #3

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 want to position the sprinkler at a point equidistant from each rosebush. Where should the garden place the sprinklers? What equation describes the boundary of the circular region that the sprinkler will cover?

Answer 1

@Jesstho.-.

Answer 2

the idea is to find the center of the circle that passes through the three points. There is a shortcut fot this, but you haven't had it yet. You could construct them to find the center, but that would be an approximation. So here goes

A(1, 3), B(5, 11), and C(11, 4)
The slopeAB = 2, so the perp slope = -1/2, and the midpoint AB is (3, 7) so y = -.5(x - 3) + 7
The slope AC = 1/10 so perp slope = -10. mdpt AC = (6, 3.5) so y = -10(x - 6) + 3.5
Combine equations to get -.5(x - 3) + 7 = -10(x - 6) + 3.5
expand: -.5x + 1.5 + 7 = -10x + 60 + 3.5
Simplify: 9.5x = 55 x = 110/19 y = 213/38

So the center of the circle, found by determining where the perpendicular bisectors of two chords, is
(110/19, 213/38)
That is where the sprinkler should be placed


Now the distance to any of the bushes is found by the distance formula. Using the point (1, 3)

sqrt(((19 - 110)/19)² + ((114 - 213)/38)²) will be the distance from the sprinkler.

sqrt(22.94+ 6.787) = sqrt(29.73) = 5.45

so the equation describing the boundary is (x - 110/19)² + (y - 213/38)² = 29.73

Answer 3

what ?

Answer 4

ohh sorry