Problem:
Rectangle PQRS lies in a plane with PQ=RS=2 and QR=SP=6. The rectangle is rotated 90β clockwise about R, then rotated 90β clockwise about the point that S moved to after the first rotation. What is the length of the path traveled by point P?
Answer Choices:
A. (23β+5β)Ο
B. 6Ο
C. (3+10β)Ο
D. (3β+25β)Ο
E. 210βΟ
Solution:
Let Pβ² and Sβ² denote the positions of P and S, respectively, after the rotation about R, and let Pβ²β² denote the final position of P. In the rotation that moves P to position Pβ², the point P rotates 90β on a circle with center R and radius PR=22+62β=210β. The length of the arc traced by P is (1/4)(2Οβ 210β)=Ο10β. Next, Pβ² rotates to Pβ²β² through a 90β arc on a circle with center Sβ² and radius Sβ²Pβ²=6. The length of this arc is 41β(2Οβ 6)=3Ο. The total distance traveled by P is