2021 AIME II Problems/Problem 12: Difference between revisions
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==Solution== | ==Solution== | ||
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 15:51, 22 March 2021
Problem
A convex quadrilateral has area 
and side lengths 
and 
in that order. Denote by 
the measure of the acute angle formed by the diagonals of the quadrilateral. Then 
can be written in the form 
, where 
and 
are relatively prime positive integers. Find 
.
Solution
Since we are asked to find , we can find 
and 
separately and then use those values to find . 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,NW);
abel("
",B,NE);
label("
",C,SE);
label("
",D,SW);
dot(X);
label("
",X,S);
label("
",(A+B)/2,N)
label("
",(B+C)/2,E);
label("
",(C+D)/2,S);
label("
",(D+A)/2,W);
label("",X,2.5E);
label("
",(A+X)/2,NE);
label("
",(B+X)/2,NW);
label("
",(C+X)/2,SW);
label("
",(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 | ||
These problems are copyrighted © by the Mathematical Association of America. 
