Difference between revisions of "2021 AIME II Problems/Problem 12"
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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== | + | ==Solution 1== |
| − | We | + | |
| + | 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>. | ||
| + | We denote by <math>E</math> the point that two diagonals <math>AC</math> and <math>BD</math> meet at. | ||
| + | To simplify the notation, we denote <math>a = AE</math>, <math>b = BE</math>, <math>c = CE</math>, <math>d = DE</math>. | ||
| + | We denote <math>\theta = \angle AED</math>. | ||
| + | |||
| + | First, we write down an equation of the area of the quadrilateral <math>ABCD</math>. | ||
| + | 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 
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 1
We denote by 
, 
, 
and 
four vertices of this quadrilateral, such that 
, 
, 
, 
.
We denote by 
the point that two diagonals 
and 
meet at.
To simplify the notation, we denote 
, 
, 
, 
.
We denote 
.
First, we write down an equation of the area of the quadrilateral 
.
We have 
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 | ||
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