Difference between revisions of "2020 AIME II Problems/Problem 4"
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After sketching, it is clear a <math>90^{\circ}</math> rotation is done about <math>(x,y)</math>. Looking between <math>A</math> and <math>A'</math>, <math>x+y=18</math> and <math>x-y=24</math>. Solving gives <math>(x,y)\implies(21,-3)</math>. Thus <math>90+21-3=\boxed{108}</math>. | After sketching, it is clear a <math>90^{\circ}</math> rotation is done about <math>(x,y)</math>. Looking between <math>A</math> and <math>A'</math>, <math>x+y=18</math> and <math>x-y=24</math>. Solving gives <math>(x,y)\implies(21,-3)</math>. Thus <math>90+21-3=\boxed{108}</math>. | ||
~mn28407 | ~mn28407 | ||
| + | |||
| + | ==Solution 2 (Official MAA)== | ||
| + | Because the rotation sends the vertical segment <math>\overline{AB}</math> to the horizontal segment <math>\overline{A'B'}</math>, the angle of rotation is <math>90^\circ</math> degrees clockwise. For any point <math>(x,y)</math> not at the origin, the line segments from <math>(0,0)</math> to <math>(x,y)</math> and from <math>(x,y)</math> to <math>(x-y,y+x)</math> are perpendicular and are the same length. Thus a <math>90^\circ</math> clockwise rotation around the point <math>(x,y)</math> sends the point <math>A(0,0)</math> to the point <math>(x-y,y+x) = A'(24,18)</math>. This has the solution <math>(x,y) = (21,-3)</math>. The requested sum is <math>90+21-3=108</math>. | ||
==Video Solution== | ==Video Solution== | ||
Revision as of 11:54, 8 June 2020
Problem
Triangles 
and 
lie in the coordinate plane with vertices 
, 
, 
, 
, 
, 
. A rotation of 
degrees clockwise around the point 
where 
, will transform to . Find 
.
Solution
After sketching, it is clear a 
rotation is done about . Looking between 
and 
, 
and 
. Solving gives 
. Thus 
.
~mn28407
Solution 2 (Official MAA)
Because the rotation sends the vertical segment 
to the horizontal segment 
, the angle of rotation is 
degrees clockwise. For any point not at the origin, the line segments from 
to and from to 
are perpendicular and are the same length. Thus a clockwise rotation around the point sends the point to the point 
. This has the solution 
. The requested sum is 
.
Video Solution
~IceMatrix
See Also=
| 2020 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
