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

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

We can't have a solution without a problem.

@@ Line 3: / Line 3: @@
 ==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.
+We can't have a solution without a problem.
-[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==
 {{AIME box|year=2021|n=II|num-b=11|num-a=13}}
 {{MAA Notice}}

2021 AIME II (Problems • Answer Key • Resources)
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Difference between revisions of "2021 AIME II Problems/Problem 12"

Revision as of 16:51, 22 March 2021

Problem

Solution

See also