Geometric rotations and reflections help?
Please explain the different rotations as well please.
- SqdancefanLv 74 years agoFavorite Answer
For the purpose of considering the effects of rotations and reflections, it might be worthwhile to think of the figure (triangle) as being drawn on a piece of (non-stretchy) transparent material that can be overlaid and moved independently from the axes of the graph.
Rotation 180° about the origin --
For this rotation, you effectively stick a pin in the overlay at the origin and rotate the figure 180°. Every (x, y) coordinate becomes a (-x, -y) coordinate after the rotation. The origin is the midpoint of every line segment connecting the original and rotated points. (It doesn't matter whether a 180° rotation is clockwise or counterclockwise, as both end in the same place.)
Rotation 90° clockwise about the origin --
Again, the pin that marks the center of rotation is stuck in the origin, but this time the overlay is rotated only 90° clockwise. If you consider the point (0, y) on the y-axis and how it moves, you find it becomes the point (y, 0) on the x-axis. Similarly, the point (x, 0) on the x-axis gets rotated to the (0, -x) position on the y-axis. Overall, the point (x, y) becomes the point (y, -x) after rotation.
Reflection across a vertical line --
Reflection across the line x=a makes the point (a, y) the midpoint of (x, y) and (x', y). (The y-coordinate remains unchanged.) In terms of our transparent overlay, we draw a line on the graph and the overlay at x=a and then flip the overlay upside down horizontally (left to right) while keeping those vertical lines on top of each other. The reflection of the point (x, y) is (2a-x, y).
Reflection across a horizontal line --
This is the same idea as reflection across a vertical line. The horizontal line on the graph and overlay at y=b remain unmoved, but the overlay is flipped over top to bottom. Point (x, y) gets translated to (x, 2b-y).
Reflection across the line y=x --
For this, we draw a diagonal line on our graph and overlay and flip over the overlay so that upper left becomes lower right and vice versa. The diagonal line we drew (y=x) remains in the same place. All coordinates (x, y) become (y, x) after the transformation.
Reflection across the line y=-x --
This is the same idea as for y=x, except our reflection line goes through the 2nd and 4th quadrants, and upper right is flipped to lower left (and vice versa). Points (x, y) become (-y, -x).
Your graph has points A', B', C' in the same clockwise order as A, B, and C. The line AB is perpendicular to the line A'B', so the transformation is not simple translation. There will either be (clockwise) rotation about a point, or there will be two reflections. (One reflection reverses the order to counterclockwise, like looking in a mirror reverses left and right.)
One can get 90° clockwise rotation from two reflections in two different ways: (1) reflection across a horizontal line followed by reflection across the line y=-x; (2) reflection across a vertical line followed by reflection across the line y=x. Checking these possibilities against the horizontal and vertical lines offered in the answer selections reveals we have ...
.. reflection across the line x=-3
.. reflection across the line y = x.
We know the figure is rotated 90° clockwise. If we assume it is about some point other than the origin, we can find that point by looking at the intersection of the perpendicular bisectors of AA', BB', and CC' (or any two of them). It turns out the center of rotation is (-3, -3). This is on both the lines y=x and x=-3.