Difference between revisions of "2022 AMC 12B Problems/Problem 5"
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~[[User: Bxiao31415 | Bxiao31415]] | ~[[User: Bxiao31415 | Bxiao31415]] | ||
− | == Solution 2 (Complex | + | == Solution 2 (Complex Numbers) == |
We write <math>(-1, -2)</math> as <math>-1-2i.</math> We'd like to rotate about <math>(3, 1),</math> which is <math>3+i</math> in the complex plane, by an angle of <math>270^{\circ} = \frac{3\pi}{2} \text{ rad}</math> counterclockwise. | We write <math>(-1, -2)</math> as <math>-1-2i.</math> We'd like to rotate about <math>(3, 1),</math> which is <math>3+i</math> in the complex plane, by an angle of <math>270^{\circ} = \frac{3\pi}{2} \text{ rad}</math> counterclockwise. | ||
The formula for rotating the complex number <math>z</math> about the complex number <math>w</math> by an angle of <math>\theta</math> counterclockwise is given as <math>(z-w)e^{\theta i} + w.</math> Plugging in our values <math>z = -1-2i, w = 3+i, \theta = \frac{3\pi}{2}</math>, we evaluate the expression as <math>((-1-2i) - (3+i))e^{\frac{3\pi i}{2}} + (3+i) = (-4-3i)(-i) + (3+i) = 4i-3i^2+3+i = 4i-3+3+i = 5i,</math> which corresponds to <math>\boxed{\textbf{(B)}\ (0,5)}</math> on the Cartesian plane. | The formula for rotating the complex number <math>z</math> about the complex number <math>w</math> by an angle of <math>\theta</math> counterclockwise is given as <math>(z-w)e^{\theta i} + w.</math> Plugging in our values <math>z = -1-2i, w = 3+i, \theta = \frac{3\pi}{2}</math>, we evaluate the expression as <math>((-1-2i) - (3+i))e^{\frac{3\pi i}{2}} + (3+i) = (-4-3i)(-i) + (3+i) = 4i-3i^2+3+i = 4i-3+3+i = 5i,</math> which corresponds to <math>\boxed{\textbf{(B)}\ (0,5)}</math> on the Cartesian plane. | ||
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+ | ~sirswagger21 | ||
==Video Solution 1== | ==Video Solution 1== |
Revision as of 20:34, 20 November 2022
Contents
Problem
The point is rotated counterclockwise about the point . What are the coordinates of its new position?
Solution 1 (Cartesian Plane)
is 4 units west and 3 units south of . Performing a counterclockwise rotation of , which is equivalent to a clockwise rotation of , the answer is 3 units west and 4 units north of , or .
Solution 2 (Complex Numbers)
We write as We'd like to rotate about which is in the complex plane, by an angle of counterclockwise.
The formula for rotating the complex number about the complex number by an angle of counterclockwise is given as Plugging in our values , we evaluate the expression as which corresponds to on the Cartesian plane.
~sirswagger21
Video Solution 1
~Education, the Study of Everything
See also
2022 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 4 |
Followed by Problem 6 |
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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.