Problem:
The point (−1,−2) is rotated 270∘ counterclockwise about the point (3,1). What are the coordinates of its new position?
Answer Choices:
A. (−3,−4)
B. (0,5)
C. (2,−1)
D. (4,3)
E. (6,−3)
Solution:
The points P(3,1),Q(−1,−2), and T(−1,1) are the vertices of a 3−4−5 triangle with right angle at T. Rotating this triangle 90∘ clockwise about P, which has the same result as rotating 270∘ counterclockwise, moves T to T′(3,5). Then the rotated image of Q is 3 units to the left at (B)(0,5).
OR
Let p=3+i and q=−1−2i be points in the complex plane corresponding to the points P and Q, respectively. Rotation about the origin by 270∘ corresponds to multiplication by −i. To rotate q an angle of 270∘ about p, the following three steps are taken.
- First, translate so that p lands on the origin of the complex plane; this sends q to q−p.
- Second, multiply q−p by −i.
- Finally, add p to the result to undo the first translation.
The resulting complex number is
(q−p)(−i)+p=((−1−2i)−(3+i))(−i)+(3+i)=(−4−3i)(−i)+(3+i)=(−3+4i)+(3+i)=0+5i.
This corresponds to the point (B)(0,5) in the coordinate plane.
OR
Translate the point (3,1) to the origin by adding the vector ⟨−3,−1⟩. Adding ⟨−3,−1⟩ to (−1,−2) produces the point (−4,−3). Rotating 90∘ clockwise about the origin, which is equivalent to rotating 270∘ counterclockwise, then moves the point (−4,−3) to (−3,4). Finally, adding the vector ⟨3,1⟩ to (−3,4) gives the requested point (B)(0,5).
The problems on this page are the property of the MAA's American Mathematics Competitions