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Difference between revisions of "2008 AMC 10A Problems/Problem 19"

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==Problem==
 
==Problem==
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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>?
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[[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>
  
 
==Solution==
 
==Solution==
−
{{solution}}
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<center><asy>
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size(220);pathpen=black+linewidth(0.65);pointpen=black;
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/* draw in rectangles */
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D(MP("R",(0,0))--MP("Q",(-6,0))--MP("P",(-6,2),N)--MP("S",(0,2),NW)--cycle);
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D((0,0)--MP("Q'",(0,6),SW)--MP("P'",(2,6),SE)--MP("S'",(2,0))--cycle);
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D(MP("R''",(2,2),NE)--MP("Q''",(8,2),N)--MP("P''",(8,0))--(2,0)--cycle);
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D(arc((0,0),(2,6),(-6,2)),dashed);D(arc((2,0),(8,0),(2,6)),dashed);D((2,6)--(0,0)--(-6,2),dashed);
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D(rightanglemark((2,6),(0,0),(-6,2),12));D(rightanglemark((2,6),(2,0),(8,0),12));
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MP("2",(-6,1),W);MP("6",(-3,0),S);
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</asy></center>
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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> and length <math>\frac14</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.
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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>.
  
 
==See also==
 
==See also==
 
{{AMC10 box|year=2008|ab=A|num-b=18|num-a=20}}
 
{{AMC10 box|year=2008|ab=A|num-b=18|num-a=20}}
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 +
[[Category:Introductory Geometry Problems]]
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{{MAA Notice}}

Latest revision as of 16:52, 7 November 2021

Problem

Rectangle $PQRS$ lies in a plane with $PQ=RS=2$ and $QR=SP=6$. The rectangle is rotated $90^\circ$ clockwise about $R$, then rotated $90^\circ$ clockwise about the point $S$ moved to after the first rotation. What is the length of the path traveled by point $P$?

$\mathrm{(A)}\ \left(2\sqrt{3}+\sqrt{5}\right)\pi\qquad\mathrm{(B)}\ 6\pi\qquad\mathrm{(C)}\ \left(3+\sqrt{10}\right)\pi\qquad\mathrm{(D)}\ \left(\sqrt{3}+2\sqrt{5}\right)\pi\\\mathrm{(E)}\ 2\sqrt{10}\pi$

Solution

[asy] size(220);pathpen=black+linewidth(0.65);pointpen=black; /* draw in rectangles */ D(MP("R",(0,0))--MP("Q",(-6,0))--MP("P",(-6,2),N)--MP("S",(0,2),NW)--cycle); D((0,0)--MP("Q'",(0,6),SW)--MP("P'",(2,6),SE)--MP("S'",(2,0))--cycle); D(MP("R''",(2,2),NE)--MP("Q''",(8,2),N)--MP("P''",(8,0))--(2,0)--cycle); D(arc((0,0),(2,6),(-6,2)),dashed);D(arc((2,0),(8,0),(2,6)),dashed);D((2,6)--(0,0)--(-6,2),dashed); D(rightanglemark((2,6),(0,0),(-6,2),12));D(rightanglemark((2,6),(2,0),(8,0),12)); MP("2",(-6,1),W);MP("6",(-3,0),S); [/asy]

We let $P'Q'R'S'$ be the rectangle after the first rotation, and $P''Q''R''S''$ be the rectangle after the second rotation. Point $P$ pivots about $R$ in an arc of a circle of radius $\sqrt{2^2+6^2} = 2\sqrt{10}$, and since $\angle PRS,\, \angle P'RQ'$ are complementary, it follows that the arc has a degree measure of $90^{\circ}$ and length $\frac14$ of the circumference. Thus, $P$ travels $\frac 14 \left(4\sqrt{10}\right)\pi = \sqrt{10}\pi$ in the first rotation.

Similarly, in the second rotation, $P$ travels in a $90^{\circ}$ arc about $S'$, with the radius being $6$. It travels $\frac 14(12)\pi = 3\pi$. Therefore, the total distance it travels is $\left(3+\sqrt{10}\right)\pi\ \mathrm{(C)}$.

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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