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?
Pt namin to pero sobrang nakakatanga.
the center of the circle will be at the intersection of the perpendicular bisectors of two of the line segments joining the points. If we label the points
A:(1,3)
B:(5,11)
C:(11,4)
then we have
AB: midpoint=(3,7) slope=2
BC: midpoint=(8,15/2) slope=-7/6
CA: midpoint=(6,7/2) slope=1/10
so, the equations of the perpendicular bisectors are
AB: y = 1/2 (17-x)
AC: y = 3/14 (4x+3)
CA: y = 1/2 (127-20x)
These all intersect at (220/38,213/38)
The radius of the circle is 5√1717/38
(x-220/38)^2 + (y-213/38)^2 = 42925/1444 = 29.726
Ouch! Hard to believe, but the math checks out.
My solutions are the same as above (Steve), but here is a step-by step guide of what needs to be done:
Step 1: The midpoint for each side of the triangle is needed, so use the Midpoint Formula with the given coordinates of the triangle.
Step 2: Find the equations that are perpendicular to at least 2 sides of the triangle using the midpoints and the Point-Slope Formula.
Step 3: Solve with Systems of Equations for the 2 equations found in Step 2. This is a challenging step! If you write the equations in Standard Form, then you can use the matrix tool in your calculator to find the ugly solutions, (110/19, 213/38). This point represents the circumcenter of the triangle, or where the sprinkler should be placed. (I think Steve forgot to reduce the fraction that is the x-coordinate.)
Step 4: Use the Distance Formula to find the length of the radius for the circle around the triangle, using the circumcenter and the points given for the vertices of the triangle. (I rounded to 5.45 .)
Step 5: Write the equation of the circle using the center point (circumcenter) and squaring the radius. ( I rounded to 29.7 .)
Good Luck! : )