The coordinates of the center of the circle passing through (1,2), (3,-4), and (5,-4)

To find the coordinates of the center of the circle passing through three given points, we can use the concept of the perpendicular bisectors of the chords. The perpendicular bisectors of the chords will intersect at the center of the circle.

Step 1: Find the midpoint of two of the given points
Let's find the midpoint of the chords formed by the points (1,2) and (3,-4), and (3,-4) and (5,-4).

Midpoint of (1,2) and (3,-4):
Midpoint x-coordinate = (1 + 3)/2 = 4/2 = 2
Midpoint y-coordinate = (2 + (-4))/2 = -2/2 = -1

Midpoint of (3,-4) and (5,-4):
Midpoint x-coordinate = (3 + 5)/2 = 8/2 = 4
Midpoint y-coordinate = (-4 + (-4))/2 = -8/2 = -4

Step 2: Find the slope of the chords
The slope of the chords passing through (1,2) and (3,-4), and (3,-4) and (5,-4) can be calculated using the formula:

Slope = (y2 - y1) / (x2 - x1)

Slope of the chord passing through (1,2) and (3,-4):
Slope = (-4 - 2) / (3 - 1) = -6 / 2 = -3

Slope of the chord passing through (3,-4) and (5,-4):
Slope = (-4 - (-4)) / (5 - 3) = 0 / 2 = 0

Step 3: Find the negative reciprocal of the slopes
To find the slope of the perpendicular bisector, we need to find the negative reciprocal of the slopes calculated in step 2.

Negative reciprocal of -3 = 1/3
Negative reciprocal of 0 = Undefined (as the slope is 0)

Step 4: Find the equation of the perpendicular bisectors
Using the midpoint and the negative reciprocals of the slopes, we can find the equations of the perpendicular bisectors.

Equation of the perpendicular bisector passing through the midpoint of (1,2) and (3,-4):
Slope = 1/3
Midpoint = (2, -1)

Using the point-slope form of the equation, we have:
y - y1 = m(x - x1)
y - (-1) = 1/3(x - 2)
y + 1 = 1/3x - 2/3
y = 1/3x - 2/3 - 1
y = 1/3x - 2/3 - 3/3
y = 1/3x - 5/3

Equation of the perpendicular bisector passing through the midpoint of (3,-4) and (5,-4):
Slope = Undefined
Midpoint = (4, -4)

Using the point-slope form of the equation, we have:
x

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