Final answer:
After a rotation of 450 degrees about the origin and then a 90-degree counter clockwise rotation about point A(3,1), point A becomes A(-1,3) and point B becomes B(1,2).
Step-by-step explanation:
The question asks for the new coordinates of points A(3,1) and B(5,2) after a rotation of 450 degrees about the origin, followed by a 90-degree counter clockwise rotation about point A. To solve this problem, we need to apply the rules of rotations in the Cartesian plane.
For the first rotation of 450°, which is equivalent to a 90° rotation (since 450° is 360° plus an additional 90°), we rotate both points 90° about the origin. The formula for a 90° rotation about the origin is (x,y) becomes (-y,x). Applying this formula:
- Point A(3,1) becomes (-1,3)
- Point B(5,2) becomes (-2,5)
Next, we rotate both points 90° counter clockwise about point A(-1,3). To do this, we first translate point A to the origin, apply the rotation, and then translate it back. The formula for a 90° counter clockwise rotation about a point (h,k) is (x,y) becomes (h-(y-k),k+(x-h)).
After applying the rotation:
- Point A remains unchanged (-1,3), since it's the pivot point.
- For Point B:
We translate B to A's origin, Apply the rotation:
B'= (-2+1,5-3) = (-1,2)
Rotate B' 90° about the origin:
B''= (-2,-1)
Translate B'' back to A's location:
Bfinal= (-1-(-2),3+(-1)) = (1,2)
So the new coordinates after all rotations are A(-1,3) and B(1,2).