Difference between revisions of "2022 AMC 12B Problems/Problem 5"
Bxiao31415 (talk | contribs) (Created page with "== Problem == The point <math>(-1, -2)</math> is rotated <math>270^{\circ}</math> counterclockwise about the point <math>(3, 1)</math>. What are the coordinates of its new pos...") |
(→Video Solution 1) |
||
(7 intermediate revisions by 5 users not shown) | |||
Line 3: | Line 3: | ||
<math>\textbf{(A)}\ (-3, -4) \qquad \textbf{(B)}\ (0,5) \qquad \textbf{(C)}\ (2,-1) \qquad \textbf{(D)}\ (4,3) \qquad \textbf{(E)}\ (6,-3)</math> | <math>\textbf{(A)}\ (-3, -4) \qquad \textbf{(B)}\ (0,5) \qquad \textbf{(C)}\ (2,-1) \qquad \textbf{(D)}\ (4,3) \qquad \textbf{(E)}\ (6,-3)</math> | ||
+ | |||
+ | == Solution 1 (Cartesian Plane) == | ||
+ | <math>(-1,-2)</math> is <math>4</math> units west and <math>3</math> units south of <math>(3,1)</math>. Performing a counterclockwise rotation of <math>270^{\circ}</math>, which is equivalent to a clockwise rotation of <math>90^{\circ}</math>, the answer is <math>3</math> units west and <math>4</math> units north of <math>(3,1)</math>, or <math>\boxed{\textbf{(B)}\ (0,5)}</math>. | ||
+ | |||
+ | ~[[User: Bxiao31415 | Bxiao31415]] | ||
+ | |||
+ | == 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. | ||
+ | |||
+ | 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. | ||
+ | |||
+ | ~sirswagger21 | ||
+ | |||
+ | ==Video Solution 1== | ||
+ | https://youtu.be/L09yN1Y5CBI | ||
+ | |||
+ | ~Education, the Study of Everything | ||
+ | |||
+ | ==Video Solution(1-16)== | ||
+ | https://youtu.be/SCwQ9jUfr0g | ||
+ | |||
+ | ~~Hayabusa1 | ||
+ | |||
+ | == See also == | ||
+ | {{AMC12 box|year=2022|ab=B|num-b=4|num-a=6}} | ||
+ | {{MAA Notice}} |
Latest revision as of 13:32, 5 April 2023
Contents
Problem
The point is rotated
counterclockwise about the point
. What are the coordinates of its new position?
Solution 1 (Cartesian Plane)
is
units west and
units south of
. Performing a counterclockwise rotation of
, which is equivalent to a clockwise rotation of
, the answer is
units west and
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
Video Solution(1-16)
~~Hayabusa1
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.