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

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(Solution 1)
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A convex quadrilateral has area <math>30</math> and side lengths <math>5, 6, 9,</math> and <math>7,</math> in that order. Denote by <math>\theta</math> the measure of the acute angle formed by the diagonals of the quadrilateral. Then <math>\tan \theta</math> can be written in the form <math>\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.
 
A convex quadrilateral has area <math>30</math> and side lengths <math>5, 6, 9,</math> and <math>7,</math> in that order. Denote by <math>\theta</math> the measure of the acute angle formed by the diagonals of the quadrilateral. Then <math>\tan \theta</math> can be written in the form <math>\tfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.
  
==Solution==
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==Solution 1==
We can't have a solution without a problem.
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We denote by <math>A</math>, <math>B</math>, <math>C</math> and <math>D</math> four vertices of this quadrilateral, such that <math>AB = 5</math>, <math>BC = 6</math>, <math>CD = 9</math>, <math>DA = 7</math>.
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We denote by <math>E</math> the point that two diagonals <math>AC</math> and <math>BD</math> meet at.
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To simplify the notation, we denote <math>a = AE</math>, <math>b = BE</math>, <math>c = CE</math>, <math>d = DE</math>.
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We denote <math>\theta = \angle AED</math>.
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First, we write down an equation of the area of the quadrilateral <math>ABCD</math>.
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We have <math>{\rm Area} \ ABCD = {\rm Area} \ \triangle ABC + </math>
  
 
==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 17:36, 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 1

We denote by $A$, $B$, $C$ and $D$ four vertices of this quadrilateral, such that $AB = 5$, $BC = 6$, $CD = 9$, $DA = 7$. We denote by $E$ the point that two diagonals $AC$ and $BD$ meet at. To simplify the notation, we denote $a = AE$, $b = BE$, $c = CE$, $d = DE$. We denote $\theta = \angle AED$.

First, we write down an equation of the area of the quadrilateral $ABCD$. We have ${\rm Area} \ ABCD = {\rm Area} \ \triangle ABC +$

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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