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Circle P has a diameter with endpoints at (–4, 3) and (1, 6). Find the center of the circle.

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5 Answers

The center of the circle is going to be at the half-way point of the diameter (or equal to the radius of the circle).
 
If you draw yourself a quick sketch of the two points you will see that the center point is equal to half the horizontal distance between the two points and half the vertical distance between the two points.
 
To work out these differences,  add up the horizontal co-ordinates and half them (X1 + X2) / 2
and do the same to the vertical co-ordinates  (Y1 + Y2) / 2
 
This will leave you with the co-ordinates of the center point of the circle.
 
  Coordinate of  a mid point of a line segment with end point coordinates of
Xm = ( X1 +X2)/ 2 ,
 Ym = (Y1 +Y2)/2.
 
      The center of a circle is midway between end points of Diameter, Therefore:
       
           Xc = ( -4 +1 ) /2 =-3/2                 Yc = ( 3 + 6) /2=9/2
 
         C : ( - 3/2, 9/2)  
 
     
 
Hello Nene -- since the center is in the middle of the ends, the center is an AVERAGE ... Xave = (1-4)/2 ... Yave = (6+3)/2 ==> C(-3/2, 9/2) ... Regards :)
On a coordinate plane you could see that the diameter of your circle is also the hypotenuse of a right triangle with endpoints at
point 1 (-4,3) and point 2 (1,6)  the lengths of the legs of that triangle are simply (x2 - x1) and (y2 - y1)
 
x2 - x1 = 1--4 = 5 is the length of the one side of that triangle, the width.
 
y2 - y1 = 6- 3 = 3 is the length of the other side of that triangle, the height.
 
 
since you want to find the midpoint of that diameter, which is your center, (xc, yc)
 
cut those lengths in half:  x length is 5/2 = 2.5 and  y height is 3/2 = 1.5 
 
add those values to the first point to arrive at the center (or you can subtract them from the 2nd point)
 
so using (-4,3) you get -4+2.5 = -1.5   xc = -1.5
 
and 3 + 1.5 = 4.5 so yc = 4.5
 
so, your answer is (-1.5,4.5) 
 
(additionally, to find the length of the diameter (hypotenuse) you use pythagoreans theorem, let D = c, where c2 = a2 + b2 and a and b are the original lengths we found above.  so, D = sqrt(52 + 32) = sqrt(34) = ~5.83  now you can use that to find the area and circumference of your circle!  woohoo!)

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and check it to see if that is a reasonable answer.  the center coordinates should fall between the original coordinates.  So,
is -1.5 between -4 and 1?  yes!   and,
is 4.5 between 3 and 6??  yes!

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The center of the circle is at the center of its diameter. Therefore, you need to find the center of the line connecting points (-4,3) and (1,6). The center of the line connecting points (x1,y1) and (x2, y2) has coordinates (½*(x1+x2);½*(y1+y2)). If you plug in the coordinates of the points given to you, you will end up with:
 
C=(½(-4+1);½(3+6))=(-3/2;9/2)
 
So, the center of the circle is at the point (-3/2;9/2)

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