For a given point (2, 0), find the coordinates of the image point under a half-turn about the origin.
I am not sure exactly what I need to do here. I already inserted a graph with the X and Y axis with the (2,0) marked. I don't know where to go from there. Please help.
For a given point (2, 0), find the coordinates of the image point under a half-turn about the origin.
I am not sure exactly what I need to do here. I already inserted a graph with the X and Y axis with the (2,0) marked. I don't know where to go from there. Please help.
For a given point (2, 0), find the coordinates of the image point under a half-turn about the origin.
I am not sure exactly what I need to do here. I already inserted a graph with the X and Y axis with the (2,0) marked. I don't know where to go from there. Please help.
If would imagines a circle with center (0,0) that goes through point (2,0), a half turn about the origin would be moving 180° (or π rad) around that circle from the given point.
First, let's parametrize our circle. Since it goes through point (2,0), we know the radius is 2 (22+02=2), and we know that a circle will be in this form:
x(θ)=rcos(θ+α) y(θ)=rsin(θ+α)
We know that r=2. To find α, we just need to solve for x(0)=2:
x(0)=2cos(α)=2→cos(α)=1, which is true at α=0
Therefore:
x(θ)=2cos(θ) y(θ)=2sin(θ)
Which satisfies (x(0),y(0))=(2,0)
A half turn would be (x(π),y(π))=(2cos(π),2sin(π))=(−2,0)
A skew or flip is about an axis; a turn (aka rotation) is around a point. To rotate around a point, the same distance is kept, but the direction (or angle) relative to the point changes. In this case, and only because the point is on the x-axis, flipping about the y-axis yields the same result as rotating one half turn around the origin. A half turn will always transform (x,y)→(−x,−y) while a y-axis flip will always transform (x,y)→(−x,y). It is clear from this why both would yield the same result at point (x1,0).
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