Hi Jesset,
When rotating a shape 180 degrees about the origin, we are moving every point 2 quadrants in the counterclockwise direction. That means that if any point lies in the first quadrant, after the rotation, it will now reside in the 3rd quadrant, any in the 2nd will move to the 4th, and so on. This means that when rotating any point about the origin by 180 degrees, the new x and y values will have the opposite sign from the original x and y values.
For example,
In the 1st quadrant, P(x,y) becomes P(-x,-y)
In the 2nd quadrant, P(-x,y) becomes P(x,-y)
and so on for the other 3rd and 4th
This question is also asking you to consider what happens when the point of rotation is different from the origin. To account for this difference, we first just have to subtract the x and y values of the point of rotation from each of the vertices before we flip the signs.
That means that when rotating triangle XYZ about the point A(5,2), for every point, we subtract 5 from the X value and 2 from the Y value,
X(5,-1) - A(5,2)
= X(5 - 5,-1 - 2)
= X(0,-3)
Y(2,2) - A(5,2)
= Y(2 - 5,2 - 2)
= Y(-3,0)
Z(0,0) - A(5,2)
= Z(0 - 5, 0 - 2)
= Z(-5,-2)
While the original points told us how far they were from the origin, the new points represent the distance each point is from the point of rotation. It is important to remember to do this step BEFORE switching the signs of the x and y values.
To apply the rotation, we then just flip the signs of all the x and y values.
X(0,-3) becomes X(0,3)
Y(-3,0) becomes Y(3,0)
Z(-5,-2) becomes Z(5,2)
Now, what we have done so far is figure out the distance each point sat from the point of rotation and then rotated those points about the ORIGIN. The last thing we need to do is add the x and y values of the point of rotation that we took out, back into the x and y values of the new points to find the final values.
X(0,3) + A(5,2)
= X'(0 + 5,3 + 2)
= X'(5,5)
Y(3,0) + A(5,2)
= Y'(3 + 5,0 + 2)
= Y'(8,2)
Z(5,2) + A(5,2)
= Z'(5 + 5,2 + 2)
= Z'(10,4)
So the final answer is
X'(5,5)
Y'(8,2)
Z'(10,4)
And that is your answer, you got it right!
The same series of steps would be followed when working with different degrees of rotation.
Hope this helps!
