Geometry

There are three points at the rim of the circle: (-4,1) (-2,-3) (5,-2). Where is the center of the lake? What is it's diameter if each unit of a coordinate plane represents 3/5 of a mile?

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  1. midpoint for (-4,1) and (-2,-3)
    = (-3,-1)
    slope through the two given points
    = -4/2 = -2
    so slope of perpendicular = 1/2
    equation of right-bisector:
    y+1 = (1/2)(x+3)
    2y + 2= x+3
    x- 2y = -1 **

    midpoint of (-4,1) and (5,-2) is (1/2 , -1/2)
    slope of chord = -3/9 = -1/3
    so slope of perpendicular = 3
    equation of right bisector:
    y + 1/2 = 3(x-1/2)
    y + 1/2 = 3x - 3/2
    2y + 1 = 6x - 3
    6x - 2y = 4 ***

    subtract equations ** and ***
    5x = 5
    x = 1
    then in **
    1-2y = -1
    -2y = -2
    y = 1

    the centre is (1,1) , and the equation is
    (x-1)^2 + (y-1)^2 = r^2
    sub in (-4,1)
    25 + 0 = r^2

    equation: (x-1)^2 + (y-1)^2 = 25
    so the radius is 5 and the diameter is 10 units on the grid, or
    10(3/5) miles = 6 miles

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