2008 AMC 10A Problems/Problem 19
Problem
Rectangle lies in a plane with and . The rectangle is rotated clockwise about , then rotated clockwise about the point moved to after the first rotation. What is the length of the path traveled by point ?
Solution
We let be the rectangle after the first rotation, and be the rectangle after the second rotation. Point pivots about in an arc of a circle of radius , and since are complementary, it follows that the arc has a degree measure of and length of the circumference. Thus, travels in the first rotation.
Similarly, in the second rotation, travels in a arc about , with the radius being . It travels . Therefore, the total distance it travels is .
See also
2008 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.