Reflect square ABCD with respect to the x-axis and the y-axis. What are the vertices of square A'B'C'D'?
To reflect a figure with respect to the x-axis, we flip the figure over the x-axis. This means that each point's y-coordinate becomes its opposite.
For square ABCD, the vertices are A(1,1), B(4,1), C(4,4), and D(1,4).
Reflecting with respect to the x-axis, the y-coordinates of each vertex become their opposites, so the new vertices are A'(1,-1), B'(4,-1), C'(4,-4), and D'(1,-4).
To reflect a figure with respect to the y-axis, we flip the figure over the y-axis. This means that each point's x-coordinate becomes its opposite.
For square ABCD, the vertices are A(1,1), B(4,1), C(4,4), and D(1,4).
Reflecting with respect to the y-axis, the x-coordinates of each vertex become their opposites, so the new vertices are A'(-1,1), B'(-4,1), C'(-4,4), and D'(-1,4).
Therefore, the vertices of square A'B'C'D' are A'(-1,1), B'(-4,1), C'(-4,4), and D'(-1,4).
To reflect a square with respect to the x-axis, we need to keep the x-coordinates of each point the same and change the sign of the y-coordinate.
Let's say that the original coordinates of the square are A(x1, y1), B(x2, y2), C(x3, y3), and D(x4, y4).
To reflect the square with respect to the x-axis, the new coordinates of the reflected square A'(x1, -y1), B'(x2, -y2), C'(x3, -y3), and D'(x4, -y4).
Similarly, to reflect a square with respect to the y-axis, we need to keep the y-coordinates of each point the same and change the sign of the x-coordinate.
So, the new coordinates of square A'B'C'D' can be obtained by reflecting the original coordinates of square ABCD with respect to both the x-axis and the y-axis.
Let's say the original square is defined as:
A(x1, y1) = (a, a)
B(x2, y2) = (a, b)
C(x3, y3) = (b, b)
D(x4, y4) = (b, a)
Reflecting square ABCD with respect to the x-axis gives us:
A'(x1, -y1) = (a, -a)
B'(x2, -y2) = (a, -b)
C'(x3, -y3) = (b, -b)
D'(x4, -y4) = (b, -a)
And reflecting square ABCD with respect to the y-axis gives us:
A"( -x1, y1) = (-a, a)
B"( -x2, y2) = (-a, b)
C"( -x3, y3) = (-b, b)
D"( -x4, y4) = (-b, a)
Therefore, the vertices of square A'B'C'D' are:
A' = (a, -a)
B' = (a, -b)
C' = (b, -b)
D' = (b, -a)