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 "2021 AIME II Problems/Problem 12"

(→‎Problem)
Line 3: Line 3:
  
 
==Solution==
 
==Solution==
−
We can't have a solution without a problem.
+
Since we are asked to find <math>\tan \theta</math>, we can find <math>\sin \theta</math> and <math>\cos \theta</math> separately and then use those values to find <math>\tan \theta</math>. Let us first draw a diagram of this quadrilateral.
 +
 
 +
[asy]
 +
unitsize(4cm);
 +
pair A,B,C,D,X;
 +
A = (0,0);
 +
B = (1,0);
 +
C = (1.25,-1);
 +
D = (-0.75,-0.75);
 +
draw(A--B--C--D--cycle,black+1bp);
 +
X = intersectionpoint(A--C,B--D);
 +
draw(A--C);
 +
draw(B--D);
 +
label("<math>A</math>",A,NW);
 +
abel("<math>B</math>",B,NE);
 +
label("<math>C</math>",C,SE);
 +
label("<math>D</math>",D,SW);
 +
dot(X);
 +
label("<math>X</math>",X,S);
 +
label("<math>5</math>",(A+B)/2,N)
 +
label("<math>6</math>",(B+C)/2,E);
 +
label("<math>9</math>",(C+D)/2,S);
 +
label("<math>7</math>",(D+A)/2,W);
 +
label("<math>\theta</math>",X,2.5E);
 +
label("<math>a</math>",(A+X)/2,NE);
 +
label("<math>b</math>",(B+X)/2,NW);
 +
label("<math>c</math>",(C+X)/2,SW);
 +
label("<math>d</math>",(D+X)/2,SE);
 +
[/asy]
 +
 
 +
~ my_aops_lessons
 +
 
  
 
==See also==
 
==See also==
 
{{AIME box|year=2021|n=II|num-b=11|num-a=13}}
 
{{AIME box|year=2021|n=II|num-b=11|num-a=13}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 16:51, 22 March 2021

Problem

A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\tan \theta$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Solution

Since we are asked to find $\tan \theta$, we can find $\sin \theta$ and $\cos \theta$ separately and then use those values to find $\tan \theta$. Let us first draw a diagram of this quadrilateral.

[asy] unitsize(4cm); pair A,B,C,D,X; A = (0,0); B = (1,0); C = (1.25,-1); D = (-0.75,-0.75); draw(A--B--C--D--cycle,black+1bp); X = intersectionpoint(A--C,B--D); draw(A--C); draw(B--D); label("$A$",A,NW); abel("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); dot(X); label("$X$",X,S); label("$5$",(A+B)/2,N) label("$6$",(B+C)/2,E); label("$9$",(C+D)/2,S); label("$7$",(D+A)/2,W); label("$\theta$",X,2.5E); label("$a$",(A+X)/2,NE); label("$b$",(B+X)/2,NW); label("$c$",(C+X)/2,SW); label("$d$",(D+X)/2,SE); [/asy]

~ my_aops_lessons


See also

2021 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
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=2021_AIME_II_Problems/Problem_12&oldid=149738"