Question 160140: Find the equation of the circle that contains the points D(7, 5), E(1, -7), and F(9, 1)
Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website! took a while but i found a solution that works and will not break your brain too badly although there are a lot of steps to get to the solution.
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if you care to do the research, the solution can be found at
http://www.regentsprep.org/Regents/math/geometry/GCG6/RCir.htm
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there's a nice picture there to show you what i mean.
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here's how i interpret that solution
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if you are given 3 points on a circle, the key to solving this is to find the intersection of 2 lines that are perpendicular to the midpoints of two lines created by the 3 points when one of the points is common to both those lines. since the 3 points are on the circle, those 2 lines are chords on the circle and intersect at the common point. the basis for this solution is that if you find the intersection of two chords on a circle, then the center of the circle is the intersection of 2 lines perpendicular to these chords at their midpoints.
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i used small letters, they used big. no change in the equations. you can make the translation yourself when you solve.
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for example:
you have 3 points (d,e,f)
they form two lines (de, ef)
these lines intersect at point e.
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you need to find the equation of a line perpendicular to the midpoint of de, and the equation of a line perpendicular to the midpoint of ef.
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once you find that, you need to find the slope of the line perpendicular to de and the slope of the line perpendicular to ef.
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once you find that you need to create the equation of the line perpendicular to de and terminating in the midpoint of de, and you need to create the equation of the line perpendicular to ef and terminating in the midpoint of ef.
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once you find that you need to find the intersection of these lines by solving the two equations simultaneously.
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once you find the intersection, you found the center of the circle.
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putting this example into the numbers of the problem you were asked to solve, we get
3 points on a circle are: d(7,5), e(1,-7), f(9,1)
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first find the midpoint of de.
the x coordinate will be (7+1)/2 = 4
the y coordinate will be (5-7)/2 = -1
the midpoint of de is (4,-1)
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next find the midpoint of ef.
the x coordinate will be (1+9)/2 = 5
the y coordinate will be ((-7)+1)/2 = -3
the midpoint of ef is (5,-3)
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reference: midpoint of de is (4,-1)
reference: midpoint of ef is (5,-3)
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next find the slope of de.
reference: general equation for slope is (y2-y1)/(x2-x1)
reference: 3 points on a circle are: d(7,5), e(1,-7), f(9,1)
slope of de is (-7-5)/1-7) = -12/-6 = 2
slope of de is 2
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next find the slope of ef.
slope of ef is (1-(-7)/9-1) = 8/8 = 1
slope of ef is 1
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reference: slope of de is 2
reference: slope of ef is 1
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next find the slope of the line perpendicular to de.
that would be (-1)/(slope of de)
which becomes -(1/2)
slope of line perpendicular to de is -(1/2)
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next find the slope of the line perpendicular to ef.
that would be (-1)/(slope of ed)
which becomes -(1)
slope of line perpendicular to ef is -(1)
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reference: slope of line perpendicular to de is -(1/2)
reference: slope of line perpendicular to ef is -(1).
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now you need to find the equation of the line perpendicular to de and terminating in the midpoint of de.
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reference: midpoint of de is (4,-1)
reference: slope of line perpendicular to de is -(1/2)
slope intercept form is y = m*x + b, where m is the slope and b is the y intercept.
this equation starts out as y-y1 = m*(x-x1) and becomes y = m*(x-x1) + y1
substituting -(1/2) for m and (4,-1) for (x1,y1) we get
y = m*(x-x1) + y1 becomes y = -(1/2)*(x-4) -1
which becomes y = -(1/2)*x -(4*(-(1/2))) - 1
which becomes y = -(1/2)*x + 2 - 1
which becomes y = (-(1/2)*x + 1
that's the slope intercept form.
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to get the standard form of the equation, you need to convert this to the form of a*x + b*y = c
multiplying both sides of the equation by 2, it becomes
2*y = (-1)*x + 2
adding (+1)*x to both sides of the equation, it becomes
x + 2*y = 2
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reference: standard form of equation for line perpendicular to de terminating in midpoint of de is x + 2*y = 2
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now you need to find the equation of the line perpendicular to ef and terminating in the midpoint of ef.
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reference: midpoint of ef is (5,-3)
reference: slope of line perpendicular to ef is -(1).
slope intercept form is y = m*x + b, where m is the slope and b is the y intercept.
this equation starts out as y-y1 = m*(x-x1) and becomes y = m*(x-x1) + y1
substituting -1 for m and (5,-3) for (x1,y1) we get
y = m*(x-x1) + y1 becomes y = -(1)*(x-5) -3
which becomes y = -(1)*x -(5*(-(1))) - 3
which becomes y = -(1)*x + 5 - 3
which becomes y = (-(1)*x + 2
that's the slope intercept form.
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to get the standard form of the equation, you need to convert this to the form of a*x + b*y = c
adding (+(1)*x to both sides of the equation, it becomes
x + y = 2
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reference: standard form of equation for line perpendicular to ef terminating in midpoint of ef is x + y = 2
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now you need to solve these 2 equations simultaneously to get their intersection.
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reference: standard form of equation for line perpendicular to de terminating in midpoint of de is x + 2*y = 2
reference: standard form of equation for line perpendicular to ef terminating in midpoint of ef is x + y = 2
simultaneous equations will be
x + 2*y = 2
x + y = 2
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solve for x in the second equation.
x = 2-y
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substitute for x in the first equation.
(2-y) + 2*y = 2
2 + y = 2
subtracting 2 from both sides of the equation gets
y = 0
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solve for x in the first equation using y = 0
x + 2*(0) = 2
x + 0 = 2
x = 2
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intersection of the line perpendicular to de and terminating in the midpoint of de, and the line perpendicular to ef and terminating in the midpoint of ef is (2,0).
this is the center of the circle.
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reference: center of circle = (2,0).
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to prove this is true, all of the radii from each of the points on the circle should be equal.
let center of circle be c.
we have 4 points: d,e,f,c
the line cd should be equal to ce should be equal to cf since these are all radii of the circle.
general form of the equation is z^2 = (x-x1)^2 + (y-y1)^2 where z^2 is the radius of the circle.
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reference: center of circle = (2,0) = c
reference: 3 points on a circle are: d(7,5), e(1,-7), f(9,1)
cd^2 = (2-7)^2 + (0-5)^2 = (-5)^2 + (-5)^2 = 25 + 25 = 50
ce^2 = (2-1)^2 + 0-(-7)^2 = (1)^2 + (7)^2 = 1 + 49 = 50
cf^2 = (2-9)^2 + (0-1)^2 = (-7)^2 + (-1)^2 = 49 + 1 = 50
all radii are equal which proves the center of the circle is correct.
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answer is:
center of circle is (2,0).
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