Difference between revisions of "2008 AMC 10A Problems/Problem 19"
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We let <math>P'Q'R'S'</math> be the first rectangle after the rotation, and <math>P''Q''R''S''</math> be the second rectangle after 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. | We let <math>P'Q'R'S'</math> be the first rectangle after the rotation, and <math>P''Q''R''S''</math> be the second rectangle after 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{( | + | 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== |
Revision as of 14:04, 23 November 2008
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
![[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]](http://latex.artofproblemsolving.com/5/b/d/5bd4d160e94d32f55888e6872d9a4748c44bab08.png)
We let be the first rectangle after the rotation, and
be the second rectangle after 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 |