Difference between revisions of "2019 AMC 12B Problems/Problem 3"
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==Problem== | ==Problem== | ||
| + | Which of the following rigid transformations (isometries) maps the line segment <math>\overline{AB}</math> onto the line segment <math>\overline{A'B'}</math> so that the image of <math>A(-2, 1)</math> is <math>A'(2, -1)</math> and the image of <math>B(-1, 4)</math> is <math>B'(1, -4)</math>? | ||
| + | |||
| + | <math>\textbf{(A) } </math> reflection in the <math>y</math>-axis | ||
| + | |||
| + | <math>\textbf{(B) } </math> counterclockwise rotation around the origin by <math>90^{\circ}</math> | ||
| + | |||
| + | <math>\textbf{(C) } </math> translation by 3 units to the right and 5 units down | ||
| + | |||
| + | <math>\textbf{(D) } </math> reflection in the <math>x</math>-axis | ||
| + | |||
| + | <math>\textbf{(E) } </math> clockwise rotation about the origin by <math>180^{\circ}</math> | ||
==Solution== | ==Solution== | ||
| + | We can simply graph the points, or use coordinate geometry, to realize that both <math>A'</math> and <math>B'</math> are, respectively, obtained by rotating <math>A</math> and <math>B</math> by <math>180^{\circ}</math> about the origin. Hence the rotation by <math>180^{\circ}</math> about the origin maps the line segment <math>\overline{AB}</math> to the line segment <math>\overline{A'B'}</math>, so the answer is <math>\boxed{(\text{E})}</math>. | ||
| + | |||
| + | ~Dodgers66 | ||
| + | |||
| + | |||
| + | ==Solution 2== | ||
| + | |||
| + | Notice that the transformation is obtained by reflecting points across the origin. Only <math>B</math> and <math>C</math> involve the origin, and since obviously reflection across the origin is <math>180^\circ</math>, the answer is <math>\boxed{(\text{E})}</math>. | ||
| + | |||
| + | ~Technodoggo | ||
| + | |||
| + | ==Video Solution 1== | ||
| + | https://youtu.be/b-htQISQ7Oo | ||
| + | |||
| + | ~Education, the Study of Everything | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2019|ab=B|num-b=2|num-a=4}} | {{AMC12 box|year=2019|ab=B|num-b=2|num-a=4}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 02:55, 24 October 2023
Problem
Which of the following rigid transformations (isometries) maps the line segment 
onto the line segment 
so that the image of 
is 
and the image of 
is 
?
reflection in the 
-axis
counterclockwise rotation around the origin by 
translation by 3 units to the right and 5 units down
reflection in the 
-axis
clockwise rotation about the origin by 
Solution
We can simply graph the points, or use coordinate geometry, to realize that both 
and 
are, respectively, obtained by rotating 
and 
by about the origin. Hence the rotation by about the origin maps the line segment to the line segment , so the answer is 
.
~Dodgers66
Solution 2
Notice that the transformation is obtained by reflecting points across the origin. Only and 
involve the origin, and since obviously reflection across the origin is 
, the answer is .
~Technodoggo
Video Solution 1
~Education, the Study of Everything
See Also
| 2019 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 2 |
Followed by Problem 4 |
| 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. 
