Difference between revisions of "2008 AMC 10A Problems/Problem 19"
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==Problem== | ==Problem== | ||
− | [[Rectangle]] <math>PQRS</math> lies in a plane with <math>PQ=RS= | + | [[Rectangle]] <math>PQRS</math> lies in a plane with <math>PQ=RS=2</math> and <math>QR=SP=6</math>. The rectangle is rotated <math>90^\circ</math> clockwise about <math>R</math>, then rotated <math>90^\circ</math> clockwise about the point <math>S</math> moved to after the first rotation. What is the length of the path traveled by point <math>P</math>? |
<math>\mathrm{(A)}\ \left(2\sqrt{3}+\sqrt{5}\right)\pi\qquad\mathrm{(B)}\ 6\pi\qquad\mathrm{(C)}\ \left(3+\sqrt{10}\right)\pi\qquad\mathrm{(D)}\ \left(\sqrt{3}+2\sqrt{5}\right)\pi\\\mathrm{(E)}\ 2\sqrt{10}\pi</math> | <math>\mathrm{(A)}\ \left(2\sqrt{3}+\sqrt{5}\right)\pi\qquad\mathrm{(B)}\ 6\pi\qquad\mathrm{(C)}\ \left(3+\sqrt{10}\right)\pi\qquad\mathrm{(D)}\ \left(\sqrt{3}+2\sqrt{5}\right)\pi\\\mathrm{(E)}\ 2\sqrt{10}\pi</math> |
Latest revision as of 16:52, 7 November 2021
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 |
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