Difference between revisions of "2016 AMC 10A Problems/Problem 16"
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+ | == Problem == | ||
+ | |||
A triangle with vertices <math>A(0, 2)</math>, <math>B(-3, 2)</math>, and <math>C(-3, 0)</math> is reflected about the <math>x</math>-axis, then the image <math>\triangle A'B'C'</math> is rotated counterclockwise about the origin by <math>90^{\circ}</math> to produce <math>\triangle A''B''C''</math>. Which of the following transformations will return <math>\triangle A''B''C''</math> to <math>\triangle ABC</math>? | A triangle with vertices <math>A(0, 2)</math>, <math>B(-3, 2)</math>, and <math>C(-3, 0)</math> is reflected about the <math>x</math>-axis, then the image <math>\triangle A'B'C'</math> is rotated counterclockwise about the origin by <math>90^{\circ}</math> to produce <math>\triangle A''B''C''</math>. Which of the following transformations will return <math>\triangle A''B''C''</math> to <math>\triangle ABC</math>? | ||
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<math>\textbf{(E)}</math> reflection about the <math>y</math>-axis. | <math>\textbf{(E)}</math> reflection about the <math>y</math>-axis. | ||
+ | |||
+ | ==Solution== | ||
+ | Consider a point <math>(x, y)</math>. Reflecting it about the <math>x</math>-axis will map it to <math>(x, -y)</math>, and rotating it counterclockwise about the origin by <math>90^{\circ}</math> will map it to <math>(y, x)</math>. The operation that undoes this is a reflection about the <math>y = x</math>, so the answer is <math>\boxed{\textbf{(D)}}</math>. | ||
+ | |||
+ | ==Video Solution (CREATIVE THINKING)== | ||
+ | https://youtu.be/xUJao7B5BHw | ||
+ | |||
+ | ~Education, the Study of Everything | ||
+ | |||
+ | |||
+ | |||
+ | ==Video Solution== | ||
+ | |||
+ | https://www.youtube.com/watch?v=VHXDQz5BG8U | ||
+ | |||
+ | ==Video Solution 2== | ||
+ | https://youtu.be/9AqQiA0aE24 | ||
+ | |||
+ | ~IceMatrix | ||
+ | |||
+ | ==See Also== | ||
+ | |||
+ | |||
+ | {{AMC10 box|year=2016|ab=A|num-b=15|num-a=17}} | ||
+ | {{MAA Notice}} |
Latest revision as of 14:52, 25 June 2023
Contents
Problem
A triangle with vertices ,
, and
is reflected about the
-axis, then the image
is rotated counterclockwise about the origin by
to produce
. Which of the following transformations will return
to
?
counterclockwise rotation about the origin by
.
clockwise rotation about the origin by
.
reflection about the
-axis
reflection about the line
reflection about the
-axis.
Solution
Consider a point . Reflecting it about the
-axis will map it to
, and rotating it counterclockwise about the origin by
will map it to
. The operation that undoes this is a reflection about the
, so the answer is
.
Video Solution (CREATIVE THINKING)
~Education, the Study of Everything
Video Solution
https://www.youtube.com/watch?v=VHXDQz5BG8U
Video Solution 2
~IceMatrix
See Also
2016 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.