Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2020 AIME II Problems/Problem 4"

(→‎Written Solution Added)
m (→‎Solution)
Line 3: Line 3:
  
 
==Solution==
 
==Solution==
−
After sketching, it is clear a <math>90^{\circ}</math> rotation is done about <math>(x,y)</math>. Looking between <math>C</math> and <math>C'</math>, <math>x+y-16=2</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
  

Revision as of 10:32, 8 June 2020

Problem

Triangles $\triangle ABC$ and $\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$, $A'(24,18)$, $B'(36,18)$, $C'(24,2)$. A rotation of $m$ degrees clockwise around the point $(x,y)$ where $0<m<180$, will transform $\triangle ABC$ to $\triangle A'B'C'$. Find $m+x+y$.

Solution

After sketching, it is clear a $90^{\circ}$ rotation is done about $(x,y)$. Looking between $A$ and $A'$, $x+y=18$ and $x-y=24$. Solving gives $(x,y)\implies(21,-3)$. Thus $90+21-3=\boxed{108}$. ~mn28407

Video Solution

https://youtu.be/atqPgGG0Ekk

~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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2020_AIME_II_Problems/Problem_4&oldid=124440"