Difference between revisions of "2021 AIME II Problems/Problem 12"
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==Video Solution== | ==Video Solution== | ||
https://www.youtube.com/watch?v=7DxIdTLNbo0 | https://www.youtube.com/watch?v=7DxIdTLNbo0 | ||
+ | |||
+ | ==Solution 4 (Symmetry)== | ||
+ | Given an inscribed quadrilateral <math>ABCD</math> with sides <math>AB = a, BC = b, CD = c,</math> and <math>DA = d.</math> Prove that the <math>\angle \theta < \frac{\pi}{2}</math> between the diagonals is given by | ||
+ | <cmath>\begin{align*}2(ac + bd) \cos \theta = {|d^2 – c^2 + b^2 – a^2|}.\end{align*}</cmath> | ||
+ | Proof: Let the point <math>B'</math> be symmetric to <math>B</math> with respect to the perpendicular bisector <math>AC.</math> Then the quadrilateral <math>AB'CD</math> is an inscribed one, <math>AB' = b, B'C = a.</math> | ||
+ | |||
+ | <cmath> 2 \angle AEB = \overset{\Large\frown} {AB} + \overset{\Large\frown} {CD}.</cmath> | ||
+ | <cmath>\begin{align*} 2\angle B'AD = \overset{\Large\frown} {B'C} + \overset{\Large\frown} {CD} = \overset{\Large\frown} {AB} + \overset{\Large\frown} {CD}.\end{align*}</cmath> | ||
+ | <cmath>\angle AEB = \angle B'AD.</cmath> | ||
+ | We apply the Law of Cosines to <math>\triangle AB'D</math> and <math>\triangle CB'D</math>: | ||
+ | <cmath>\begin{align*} B'D^2 = AD^2 + AB'^2 – 2 AD \cdot AB' \cos \theta = CD^2 + CB'^2 + 2 CD \cdot CB' \cos \theta,\end{align*}</cmath> | ||
+ | <cmath>\begin{align*} d^2 + b^2 – 2 b\cdot d \cos \theta = c^2 + a^2 + 2a\cdot c \cos \theta,\end{align*}</cmath> | ||
+ | <cmath>\begin{align*} 2(ac + bd) \cos \theta = |d^2 – c^2 + b^2 – a^2|.\end{align*}</cmath> | ||
+ | |||
+ | [[File:2021 AIME I 11c.jpg]] | ||
+ | |||
+ | ~vvsss, www.deoma-cmd.ru | ||
+ | |||
+ | |||
+ | |||
+ | |||
==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 01:41, 30 May 2022
Contents
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 (Sines and Cosines)
Since we are asked to find , we can find
and
separately and use their values to get
. We can start by drawing a diagram. Let the vertices of the quadrilateral be
,
,
, and
. Let
,
,
, and
. Let
,
,
, and
. We know that
is the acute angle formed between the intersection of the diagonals
and
.
We are given that the area of quadrilateral
is
. We can express this area using the areas of triangles
,
,
, and
. Since we want to find
and
, we can represent these areas using
as follows:
We know that
. Therefore it follows that:
From here we see that
. Now we need to find
. Using the Law of Cosines on each of the four smaller triangles, we get following equations:
We know that
for all
. We can substitute this value into our equations to get:
If we subtract
from
, the squared terms cancel, leaving us with:
From here we see that
.
Since we have figured out and
, we can calculate
:
Therefore our answer is
.
~ Steven Chen (www.professorchenedu.com)
~ my_aops_lessons
Solution 2 (Right Triangles)
In convex quadrilateral let
and
Let
and
be the feet of the perpendiculars from
and
respectively, to
We obtain the following diagram:
Let
and
We apply the Pythagorean Theorem to right triangles
and
respectively:
Let the brackets denote areas. We get
We subtract
from
From right triangles
and
we have
It follows that
Finally, we divide
by
from which the answer is
~MRENTHUSIASM
Solution 3 (Bretschneider's Formula)
Bretschneider's Formula
Given quadrilateral
, let,
, be the sides,
the semiperimeter, and
, the diagonals. Then the area,
, is given by
Solution
By Bretschneider's Formula, Thus,
. Also,
solving for
yields
. Since
is acute,
is positive, from which
. Solving for
yields
for a final answer of
.
~ Leo.Euler
Video Solution
https://www.youtube.com/watch?v=7DxIdTLNbo0
Solution 4 (Symmetry)
Given an inscribed quadrilateral with sides
and
Prove that the
between the diagonals is given by
Proof: Let the point
be symmetric to
with respect to the perpendicular bisector
Then the quadrilateral
is an inscribed one,
We apply the Law of Cosines to
and
:
~vvsss, www.deoma-cmd.ru
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.