2021 AIME II Problems/Problem 12
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 use the triangle area formula with sines to write down an equation of the area of the quadrilateral .
We have
.
Because , we have
. We index this equation as Eq (1).
Second, we use the law of cosines to establish four equations for four sides of the quadrilateral .
By applying the law of cosines to , we have
.
Note that
.
Hence,
.
We index this equation as Eq (2).
Analogously, we can establish the following equation for that
(indexed as Eq (3)), the following equation for
that
(indexed as Eq (4)) and the following equation for
that
.
~ Steven Chen (from Professor Chen Education Palace)
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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