the end points of the diameter of the circle are(2,-1)and (5,4).find the coordinates of its centre
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briefcaseTeacher (K-12)
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The center of a circle is at the midpoint of any of its diameters. In particular, the center of the circle in question will be at the midpoint of the segment joining (2,-1) and (5,4).
The midpoint of a segment is found by averaging the x-coordinates and the y-coordinates.
The midpoint, and thus the center of the circle, is at ((2+5)/2,(-1+4)/2) or (7/2,3/2).
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Any diameter of a circle is going to go through the center point of the circle. Perhaps even more importantly, the midpoint of the diameter will be where the center of the circle is. This makes intuitive sense because any distance from the center of the circle to a point on the circle is equal to the radius length. The diameter is twice the length of the radius, so it makes sense that the center point of the circle evenly bisects the diameter.
Applying these concepts to the problem on hand then means that to find the center of the circle, all we have to do is find the midpoint of the diameter using the endpoints were are given. To find the middle point of a line segment given the line segment's endpoints, all we have to do is find the average of the x-coordinates and the average of the y-coordinates. The averages will then provide the coordinates of the midpoint.
Average of the x-coordinates = (2 + 5) / 2 = 3.5
Average of the y-coordinates = (-1 + 4) / 2 = 1.5
Therefore, the center point is found at (3.5, 1.5)
To find a midpoint, or center, you simply find the average of your x-values and y-values. An average is found by adding the numbers together and dividing by how many numbers you have (in this case it'd be 2).
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For your midpoints x-value, you will average the x-values of the 2 given points.
(2+5)/2 = 7/2
For your midpoints y-value, you will average the y-values of the 2 given points.
(-1+4)/2 = 3/2
This makes your midpoint (7/2,3/2)
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