Question 985218
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A perpendicular bisector of a chord of a circle passes through the center of the circle.  Perpendicular bisectors of two different chords on the same circle will intersect in the center of the circle.  The distance from the center to any of the endpoints of a chord is the radius of the circle.


1.  Use the slope formula to calculate the slope of the line through one pair of your given points.


2.  Use the midpoint formulas to calculate the coordinates of the midpoint of the line segment that joins the two points you selected for step 1.


3.  Calculate the negative reciprocal of the slope you calculated in step 1.


4.  Use the point-slope form of an equation of a line with the slope calculated in step 3 and the point determined in step 2 to derive the equation of the perpendicular bisector of the chord that joins the two points selected for step 1.


5.  Repeat steps 1 through 4 for a different pair of given points.


6.  The equations derived in steps 4 and 5 form a 2X2 linear system.  Solve this system for the coordinates of the point of intersection.  This will be the center of circle.  For the moment, consider this point to be *[tex \Large (h,k)]


7.  Use the distance formula to determine the distance between *[tex \Large (h,k)] and one of your given points.  This will be *[tex \Large r], the radius of the circle.


8.  Write the equation of the circle using the coordinates of the center and the value of the radius:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ (x\ -\ h)^2\ +\ (y\ -\ k)^2\ =\ r^2]


<b>References</b>


Slope Formula:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ m\ =\ \frac{y_1\ -\ y_2}{x_1\ -\ x_2} ]


where *[tex \Large \left(x_1,y_1\right)] and *[tex \Large \left(x_2,y_2\right)] are the coordinates of the given points.


Slope relationship between perpendicular lines:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  L_1\ \perp\ L_2 \ \ \Leftrightarrow\ \ m_1\ =\ -\frac{1}{m_2}\ \text{ and } m_1,\, m_2\, \not =\, 0]


Midpoint formulas:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x_m\ = \frac{x_1 + x_2}{2}] and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y_m\ = \frac{y_1 + y_2}{2}]


Point-slope form:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ -\ y_1\ =\ m(x\ -\ x_1) ]


where *[tex \Large \left(x_1,y_1\right)] are the coordinates of the given point and *[tex \Large m] is the calculated slope.


Distance Formula:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ d\ =\ \sqrt{(x_1\ -\ x_2)^2\ +\ (y_1\ -\ y_2)^2}]


*[illustration Circ3Points1].


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it

*[tex \Large \ \
*[tex \LARGE \ \ \ \ \ \ \ \ \ \