Difference between revisions of "2008 AMC 10A Problems/Problem 19"
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− | We let <math>P'Q'R'S'</math> be the | + | We let <math>P'Q'R'S'</math> be the rectangle after the first rotation, and <math>P''Q''R''S''</math> be the rectangle after the second rotation. Point <math>P</math> pivots about <math>R</math> in an [[arc]] of a circle of radius <math>\sqrt{2^2+6^2} = 2\sqrt{10}</math>, and since <math>\angle PRS,\, \angle P'RQ'</math> are complementary, it follows that the arc has a degree measure of <math>90^{\circ}</math> (or <math>1/4</math> of the [[circumference]]). Thus, <math>P</math> travels <math>\frac 14 \left(4\sqrt{10}\right)\pi = \sqrt{10}\pi</math> in the first rotation. |
Similarly, in the second rotation, <math>P</math> travels in a <math>90^{\circ}</math> arc about <math>S'</math>, with the radius being <math>6</math>. It travels <math>\frac 14(12)\pi = 3\pi</math>. Therefore, the total distance it travels is <math>\left(3+\sqrt{10}\right)\pi\ \mathrm{(C)}</math>. | Similarly, in the second rotation, <math>P</math> travels in a <math>90^{\circ}</math> arc about <math>S'</math>, with the radius being <math>6</math>. It travels <math>\frac 14(12)\pi = 3\pi</math>. Therefore, the total distance it travels is <math>\left(3+\sqrt{10}\right)\pi\ \mathrm{(C)}</math>. |
Revision as of 18:31, 4 June 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 (or 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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