Question 262337
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If you know the endpoints of the diameter of a circle, you can use the midpoint formulas to calculate the midpoint of the diameter which is the center of the circle.  Then the distance formula can be used to calculate the distance from the center to either endpoint of the diameter which is equal to the radius of the circle.


The *[tex \Large x]-coordinate of the mid-point of the diameter segment, and therefore the *[tex \Large x]-coordinate of the center of the circle is given by:


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


and the *[tex \Large y]-coordinate of the mid-point of the diameter segment, and therefore the *[tex \Large y]-coordinate of the center of the circle is given by:


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


Where *[tex \Large \left(x_1,y_1\right)] and *[tex \Large \left(x_2,y_2\right)] are the coordinates of the endpoints of the diameter.


The measure of the segment from the center of the circle, *[tex \Large \left(h,\,k\right)], to either endpoint of the diameter, either *[tex \Large \left(x_1,y_1\right)] or *[tex \Large \left(x_2,y_2\right)] is given by the distance formula:


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


(or *[tex \LARGE d\ =\ sqrt{(h\ -\ x_2)^2\ +\ (k\ -\ y_2)^2}] if you prefer)


To graph, set your compass to the calculated radius and strike a full-circle arc with the compass point at the calculated center, *[tex \Large \left(h,\,k\right)]


The equation of a circle centered at *[tex \Large \left(h,\,k\right)] and with radius *[tex \Large r] is:


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


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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