Difference between revisions of "2021 AIME II Problems/Problem 12"

(Problem)
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==Solution==
 
==Solution==
We can't have a solution without a problem.
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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;
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A = (0,0);
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B = (1,0);
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C = (1.25,-1);
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D = (-0.75,-0.75);
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draw(A--B--C--D--cycle,black+1bp);
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X = intersectionpoint(A--C,B--D);
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draw(A--C);
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draw(B--D);
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label("<math>A</math>",A,NW);
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abel("<math>B</math>",B,NE);
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label("<math>C</math>",C,SE);
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label("<math>D</math>",D,SW);
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dot(X);
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label("<math>X</math>",X,S);
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label("<math>5</math>",(A+B)/2,N)
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label("<math>6</math>",(B+C)/2,E);
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label("<math>9</math>",(C+D)/2,S);
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label("<math>7</math>",(D+A)/2,W);
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label("<math>\theta</math>",X,2.5E);
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label("<math>a</math>",(A+X)/2,NE);
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label("<math>b</math>",(B+X)/2,NW);
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label("<math>c</math>",(C+X)/2,SW);
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label("<math>d</math>",(D+X)/2,SE);
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[/asy]
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 +
~ 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 (ProblemsAnswer KeyResources)
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

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