In figure LMNO below, segments MN and LO are parallel. The segments lie on the lines y = 6 and y = 3.
If the figure LMNO is rotated 90° clockwise about the origin to create figure L'M'N'O', what lines will the corresponding parallel segments lie on?
When the figure is rotated 90° clockwise about the origin, the x-coordinate of each point becomes its y-coordinate and the y-coordinate of each point becomes the negative of its x-coordinate.
In this case, segment MN lies on the line y = 6, so its endpoint M has coordinates (m, 6) and its endpoint N has coordinates (n, 6).
When the figure is rotated 90° clockwise, M' will have coordinates (6, -m) and N' will have coordinates (6, -n). Therefore, segment M'N' lies on the line x = 6.
Similarly, segment LO lies on the line y = 3, so its endpoint L has coordinates (l, 3) and its endpoint O has coordinates (o, 3).
When the figure is rotated 90° clockwise, L' will have coordinates (3, -l) and O' will have coordinates (3, -o). Therefore, segment L'O' lies on the line x = 3.
So the corresponding parallel segments in figure L'M'N'O' will lie on the lines x = 3 and x = 6.
To find out the lines on which the corresponding parallel segments will lie after the clockwise rotation of figure LMNO, we need to understand the effect of a 90° clockwise rotation on the given lines.
A 90° clockwise rotation about the origin transforms a point (x, y) to (-y, x). Using this transformation, we can determine the new equations of the lines.
1. Line y = 6:
After rotating 90° clockwise, the new equation will be x = -6.
Therefore, the corresponding parallel segment will lie on the line x = -6.
2. Line y = 3:
After rotating 90° clockwise, the new equation will be x = -3.
Therefore, the corresponding parallel segment will lie on the line x = -3.
So, the corresponding parallel segments in figure L'M'N'O' will lie on the lines x = -6 and x = -3, respectively.
To find the lines that the corresponding parallel segments will lie on after a 90° clockwise rotation, we need to first understand the effect of a 90° clockwise rotation on the coordinate axes.
When a point (x, y) is rotated 90° clockwise about the origin (0, 0), the new coordinates (x', y') are given by:
x' = y
y' = -x
Now let's apply this transformation to the given lines and see how they change.
The first line, y = 6, has a constant y-coordinate of 6. Applying the rotation transformation, we get:
x' = 6 (since y = 6 remains the same)
y' = -x
So, the equation of the first line after the 90° clockwise rotation will be x' = 6.
Similarly, for the second line, y = 3, we get:
x' = 3 (since y = 3 remains the same)
y' = -x
Therefore, the equation of the second line after the 90° clockwise rotation will be x' = 3.
Hence, the corresponding parallel segments in the rotated figure L'M'N'O' will lie on the lines x' = 6 and x' = 3, respectively.