Points A and B have symmetry with respect to point C.

Find the coordinates of C given the points:
a) A(3, 4) and B(5, 1) c) A(5, 3) and B(2, 1)
b) A(0, 2) and B(0, 6) d) A(2a, 0) and B(0, 2b)

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a. A(3,4), C(x,y), B(5,1).

Points A and B are equidistance from C:
x-3 = 5-x
X = 4

y-4 = 1-y
Y = 2 1/2

b. A(0,2), C(x,y), B(0,6).
x-0 = 0-x
X = 0

y-2 = 6-y
Y = 4

c. Same procedure as a, and b.

d. A(2a,0), C(x,y), B(0,2b).
x-2a = 0-x
2x = 2a
X = a

y-0 = 2b-y
2y = 2b
Y = b

Another Method:

a. X = (3+5)/2 = 4
Y = (4+1)/2 = 2.5

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To find the coordinates of C when points A and B have symmetry with respect to point C, you can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint of a line segment AB, in this case, are given by the averages of the corresponding coordinates of A and B.

a) A(3, 4) and B(5, 1):
To find the coordinates of C, the midpoint of AB, add the x-coordinates and divide by 2 to get the x-coordinate of C:
x-coordinate of C = (3 + 5) / 2 = 8 / 2 = 4

Do the same for the y-coordinates to get the y-coordinate of C:
y-coordinate of C = (4 + 1) / 2 = 5 / 2 = 2.5

Therefore, the coordinates of C are (4, 2.5).

c) A(5, 3) and B(2, 1):
Using the same method, the x-coordinate of C is (5 + 2) / 2 = 7 / 2 = 3.5
And the y-coordinate of C is (3 + 1) / 2 = 4 / 2 = 2

Therefore, the coordinates of C are (3.5, 2).

b) A(0, 2) and B(0, 6):
For this case, the x-coordinate of A and B is the same (0). To find the x-coordinate of C, the x-coordinate of the midpoint remains 0.

For the y-coordinate, it would be (2 + 6) / 2 = 8 / 2 = 4

Therefore, the coordinates of C are (0, 4).

d) A(2a, 0) and B(0, 2b):
In this case, the x-coordinate of A is 2a and the x-coordinate of B is 0. To find the x-coordinate of C, calculate the average of these two values:
x-coordinate of C = (2a + 0) / 2 = 2a / 2 = a

For the y-coordinate, the y-coordinate of A is 0, and the y-coordinate of B is 2b. Calculate the average of these two values:
y-coordinate of C = (0 + 2b) / 2 = 2b / 2 = b

Therefore, the coordinates of C are (a, b).