Difference between revisions of "2022 AIME II Problems/Problem 11"

(Solution)
(Solution 1)
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==Problem==
 
==Problem==
Let <math>ABCD</math> be a convex quadrilateral with <math>AB=2</math>, <math>AD=7</math>, and <math>CD=3</math> such that the bisectors of acute angles <math>\angle{DAB}</math> and <math>\angle{ADC}</math> intersect at the midpoint of <math>\overline{BC}</math>. Find the square of the area of <math>ABCD</math>.
 
  
==Solution==
+
Let <math>ABCD</math> be a convex quadrilateral with <math>AB=2, AD=7,</math> and <math>CD=3</math> such that the bisectors of acute angles <math>\angle{DAB}</math> and <math>\angle{ADC}</math> intersect at the midpoint of <math>\overline{BC}.</math> Find the square of the area of <math>ABCD.</math>
  
[[Image:2022AIME2-Q11.png|thumb|center|500px|2022 AIME II Q11(Hand-draw picture)]]
+
==Solution 1==
  
 +
<asy>
 +
defaultpen(fontsize(12)+0.6); size(300);
 +
pair A,B,C,D,M,H; real xb=71, xd=121;
 +
A=origin; D=(7,0); B=2*dir(xb); C=3*dir(xd)+D; M=(B+C)/2; H=foot(M,A,D); path c=CR(D,3); pair A1=bisectorpoint(D,A,B), D1=bisectorpoint(C,D,A), Bp=IP(CR(A,2),A--H), Cp=IP(CR(D,3),D--H);
 +
 +
draw(B--A--D--C--B); draw(A--M--D^^M--H^^Bp--M--Cp, gray+0.4); draw(rightanglemark(A,H,M,5));
 +
 +
dot("$A$",A,SW); dot("$D$",D,SE); dot("$B$",B,NW); dot("$C$",C,NE); dot("$M$",M,up); dot("$H$",H,down); dot("$B'$",Bp,down); dot("$C'$",Cp,down);
 +
</asy>
  
 
According to the problem, we have <math>AB=AB'=2</math>, <math>DC=DC'=3</math>, <math>MB=MB'</math>, <math>MC=MC'</math>, and <math>B'C'=7-2-3=2</math>
 
According to the problem, we have <math>AB=AB'=2</math>, <math>DC=DC'=3</math>, <math>MB=MB'</math>, <math>MC=MC'</math>, and <math>B'C'=7-2-3=2</math>
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This is when we found that points <math>M</math>, <math>C</math>, <math>D</math>, and <math>B'</math> are on a circle. Thus, <math>\angle{BMB'}=\angle{CDC'} \Rightarrow \angle{B'MA}=\angle{C'DM}</math>. This is the time we found that <math>\bigtriangleup{AB'M} \sim \bigtriangleup{MC'D}</math>.
 
This is when we found that points <math>M</math>, <math>C</math>, <math>D</math>, and <math>B'</math> are on a circle. Thus, <math>\angle{BMB'}=\angle{CDC'} \Rightarrow \angle{B'MA}=\angle{C'DM}</math>. This is the time we found that <math>\bigtriangleup{AB'M} \sim \bigtriangleup{MC'D}</math>.
  
Thus, <math>\frac{AB'}{B'M}=\frac{MC'}{C'D} \Longrightarrow (BM')^2=AB' \cdot C'D = 6</math>
+
Thus, <math>\frac{AB'}{B'M}=\frac{MC'}{C'D} \Longrightarrow (B'M)^2=AB' \cdot C'D = 6</math>
  
 
Point <math>H</math> is the midpoint of <math>B'C'</math>, and <math>MH \perp AD</math>. <math>B'H=HC'=1 \Longrightarrow MH=\sqrt{B'M^2-B'H^2}=\sqrt{6-1}=\sqrt{5}</math>.
 
Point <math>H</math> is the midpoint of <math>B'C'</math>, and <math>MH \perp AD</math>. <math>B'H=HC'=1 \Longrightarrow MH=\sqrt{B'M^2-B'H^2}=\sqrt{6-1}=\sqrt{5}</math>.
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Finally, the square of the area is <math>(6\sqrt{5})^2=\boxed{180}</math>
 
Finally, the square of the area is <math>(6\sqrt{5})^2=\boxed{180}</math>
  
~DSAERF-CALMIT
+
==Solution 2==
 +
 
 +
Denote by <math>M</math> the midpoint of segment <math>BC</math>.
 +
Let points <math>P</math> and <math>Q</math> be on segment <math>AD</math>, such that <math>AP = AB</math> and <math>DQ = DC</math>.
 +
 
 +
Denote <math>\angle DAM = \alpha</math>, <math>\angle BAD = \beta</math>, <math>\angle BMA = \theta</math>, <math>\angle CMD = \phi</math>.
 +
 
 +
Denote <math>BM = x</math>. Because <math>M</math> is the midpoint of <math>BC</math>, <math>CM = x</math>.
 +
 
 +
Because <math>AM</math> is the angle bisector of <math>\angle BAD</math> and <math>AB = AP</math>, <math>\triangle BAM \cong \triangle PAM</math>.
 +
Hence, <math>MP = MB</math> and <math>\angle AMP = \theta</math>.
 +
Hence, <math>\angle MPD = \angle MAP + \angle PMA = \alpha + \theta</math>.
 +
 
 +
Because <math>DM</math> is the angle bisector of <math>\angle CDA</math> and <math>DC = DQ</math>, <math>\triangle CDM \cong \triangle QDM</math>.
 +
Hence, <math>MQ = MC</math> and <math>\angle DMQ = \phi</math>.
 +
Hence, <math>\angle MQA = \angle MDQ + \angle QMD = \beta + \phi</math>.
 +
 
 +
Because <math>M</math> is the midpoint of segment <math>BC</math>, <math>MB = MC</math>.
 +
Because <math>MP = MB</math> and <math>MQ = MC</math>, <math>MP = MQ</math>.
 +
 
 +
Thus, <math>\angle MPD = \angle MQA</math>.
 +
 
 +
Thus,
 +
<cmath>\[
 +
\alpha + \theta = \beta + \phi . \hspace{1cm} (1)
 +
\]</cmath>
 +
 
 +
In <math>\triangle AMD</math>, <math>\angle AMD = 180^\circ - \angle MAD - \angle MDA = 180^\circ - \alpha - \beta</math>.
 +
In addition, <math>\angle AMD = 180^\circ - \angle BMA - \angle CMD = 180^\circ - \theta - \phi</math>.
 +
Thus,
 +
<cmath>\[
 +
\alpha + \beta = \theta + \phi . \hspace{1cm} (2)
 +
\]</cmath>
 +
 
 +
Taking <math>(1) + (2)</math>, we get <math>\alpha = \phi</math>.
 +
Taking <math>(1) - (2)</math>, we get <math>\beta = \theta</math>.
 +
 
 +
Therefore, <math>\triangle ADM \sim \triangle AMB \sim \triangle MDC</math>.
 +
 
 +
Hence, <math>\frac{AD}{AM} = \frac{AM}{AB}</math> and <math>\frac{AD}{DM} = \frac{DM}{CD}</math>.
 +
Thus, <math>AM = \sqrt{AD \cdot AD} = \sqrt{14}</math> and <math>DM = \sqrt{AD \cdot CD} = \sqrt{21}</math>.
 +
 
 +
In <math>\triangle ADM</math>, by applying the law of cosines, <math>\cos \angle AMD  = \frac{AM^2 + DM^2 - AD^2}{2 AM \cdot DM} = - \frac{1}{\sqrt{6}}</math>.
 +
Hence, <math>\sin \angle AMD = \sqrt{1 - \cos^2 \angle AMD} = \frac{\sqrt{5}}{\sqrt{6}}</math>.
 +
Hence, <math>{\rm Area} \ \triangle ADM = \frac{1}{2} AM \cdot DM \dot \sin \angle AMD = \frac{7 \sqrt{5}}{2}</math>.
 +
 
 +
Therefore,
 +
<cmath>\begin{align*}
 +
{\rm Area} \ ABCD & = {\rm Area} \ \triangle AMD + {\rm Area} \ \triangle ABM + {\rm Area} \ \triangle MCD \\
 +
& = {\rm Area} \ \triangle AMD \left( 1 + \left( \frac{AM}{AD} \right)^2 + \left( \frac{MD}{AD} \right)^2 \right) \\
 +
& = 6 \sqrt{5} .
 +
\end{align*}</cmath>
 +
 
 +
Therefore, the square of <math>{\rm Area} \ ABCD</math> is <math>\left( 6 \sqrt{5} \right)^2 = \boxed{\textbf{(180) }}</math>.
 +
 
 +
~Steven Chen (www.professorchenedu.com)
 +
 
 +
==Solution 3 (Visual)==
 +
[[File:AIME-II-2022-11a.png|300px|right]]
 +
<b><i>Claim</b></i>
 +
 
 +
In the triangle <math>ABC, AB = 2AC, M</math> is the midpoint of <math>AB. D</math> is the point of intersection of the circumcircle and the bisector of angle <math>A.</math> Then <math>DM = BD.</math>
 +
 
 +
<b><i> Proof</b></i>
 +
 
 +
Let <math>A = 2\alpha.</math> Then <math>\angle DBC = \angle DCB = \alpha.</math>
 +
 
 +
Let <math>E</math> be the intersection point of the perpendicular dropped from <math>D</math> to <math>AB</math> with the circle.
 +
 
 +
Then the sum of arcs  <math>\overset{\Large\frown} {BE} + \overset{\Large\frown}{AC} + \overset{\Large\frown}{CD} = 180^\circ.</math>
 +
<cmath>\overset{\Large\frown} {BE} = 180^\circ – 2\alpha – \overset{\Large\frown}{AC}.</cmath>
 +
 
 +
Let <math>E'</math> be the point of intersection of the line <math>CM</math> with the circle.
 +
<math>CM</math> is perpendicular to <math>AD, \angle AMC = 90^\circ – \alpha,</math> the sum of arcs  <math>\overset{\Large\frown}{A}C +  \overset{\Large\frown}{BE'} = 180^\circ – 2\alpha \implies E'</math> coincides with <math>E.</math>
 +
 
 +
The inscribed angles <math>\angle DEM = \angle DEB, M</math> is symmetric to <math>B</math> with respect to <math>DE, DM = DB.</math>
 +
 
 +
<b><i> Solution</b></i>
 +
[[File:AIME-II-2022-11b.png|350px|right]]
 +
Let <math>AB' = AB, DC' = DC, B'</math> and <math>C'</math> on <math>AD.</math>
 +
 
 +
Then <math>AB' = 2, DC' = 3, B'C' = 2 = AB'.</math>
 +
 
 +
Quadrilateral  <math>ABMC'</math> is cyclic.
 +
Let <math>\angle A = 2\alpha.</math> Then <math>\angle MBC' = \angle MC'B = \alpha.</math>
 +
 
 +
Circle <math>BB'C'C</math> centered at <math>M, BC</math> is its diameter, <math>\angle BC'C = 90^\circ.</math>
 +
<math>\angle DMC' = \angle MC'B,</math> since they both complete <math>\angle MC'C</math> to <math>90^\circ.</math>
 +
 
 +
<math>\angle MB'A = \angle MC'D,</math> since they are the exterior angles of an isosceles <math>\triangle MB'C'.</math>
 +
<math>\triangle AMB' \sim \triangle MDC'</math> by two angles.
 +
<math>\frac {AB'}{MC'} = \frac {MB'}{DC'}, MC' =\sqrt{AB' \cdot C'D} = \sqrt{6}.</math>
 +
 
 +
The height dropped from <math>M</math> to <math>AD</math> is <math>\sqrt{MB'^2 - (\frac{B'C'}{2})^2} =\sqrt{6 - 1} = \sqrt{5}.</math>
 +
 
 +
The areas of triangles <math>\triangle AMB'</math> and <math>\triangle MC'B'</math> are equal to <math>\sqrt{5},</math> area of <math>\triangle MC'D</math> is <math>\frac{3}{2} \sqrt{5}.</math>
 +
 
 +
<cmath>\triangle AMB' = \triangle AMB, \triangle MC'D = \triangle MCD \implies</cmath>
 +
The area of  <math>ABCD</math> is <math>(1 + 2 + 3) \sqrt{5} = 6\sqrt{5} \implies 6^2 \cdot 5 = \boxed{180}.</math>
 +
 
 +
'''vladimir.shelomovskii@gmail.com, vvsss'''
 +
 
 +
==Video Solution by The Power of Logic==
 +
https://youtu.be/giLyWHKFr1I
  
 
==See Also==
 
==See Also==
 
{{AIME box|year=2022|n=II|num-b=10|num-a=12}}
 
{{AIME box|year=2022|n=II|num-b=10|num-a=12}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 08:18, 30 March 2023

Problem

Let $ABCD$ be a convex quadrilateral with $AB=2, AD=7,$ and $CD=3$ such that the bisectors of acute angles $\angle{DAB}$ and $\angle{ADC}$ intersect at the midpoint of $\overline{BC}.$ Find the square of the area of $ABCD.$

Solution 1

[asy]  defaultpen(fontsize(12)+0.6); size(300);  pair A,B,C,D,M,H; real xb=71, xd=121; A=origin; D=(7,0); B=2*dir(xb); C=3*dir(xd)+D; M=(B+C)/2; H=foot(M,A,D); path c=CR(D,3); pair A1=bisectorpoint(D,A,B), D1=bisectorpoint(C,D,A), Bp=IP(CR(A,2),A--H), Cp=IP(CR(D,3),D--H);  draw(B--A--D--C--B); draw(A--M--D^^M--H^^Bp--M--Cp, gray+0.4); draw(rightanglemark(A,H,M,5));  dot("$A$",A,SW); dot("$D$",D,SE); dot("$B$",B,NW); dot("$C$",C,NE); dot("$M$",M,up); dot("$H$",H,down); dot("$B'$",Bp,down); dot("$C'$",Cp,down); [/asy]

According to the problem, we have $AB=AB'=2$, $DC=DC'=3$, $MB=MB'$, $MC=MC'$, and $B'C'=7-2-3=2$

Because $M$ is the midpoint of $BC$, we have $BM=MC$, so: \[MB=MB'=MC'=MC.\]

Then, we can see that $\bigtriangleup{MB'C'}$ is an isosceles triangle with $MB'=MC'$

Therefore, we could start our angle chasing: $\angle{MB'C'}=\angle{MC'B'}=180^\circ-\angle{MC'D}=180^\circ-\angle{MCD}$.

This is when we found that points $M$, $C$, $D$, and $B'$ are on a circle. Thus, $\angle{BMB'}=\angle{CDC'} \Rightarrow \angle{B'MA}=\angle{C'DM}$. This is the time we found that $\bigtriangleup{AB'M} \sim \bigtriangleup{MC'D}$.

Thus, $\frac{AB'}{B'M}=\frac{MC'}{C'D} \Longrightarrow (B'M)^2=AB' \cdot C'D = 6$

Point $H$ is the midpoint of $B'C'$, and $MH \perp AD$. $B'H=HC'=1 \Longrightarrow MH=\sqrt{B'M^2-B'H^2}=\sqrt{6-1}=\sqrt{5}$.

The area of this quadrilateral is the sum of areas of triangles: \[S_{\bigtriangleup{ABM}}+S_{\bigtriangleup{AB'M}}+S_{\bigtriangleup{CDM}}+S_{\bigtriangleup{CD'M}}+S_{\bigtriangleup{B'C'M}}\] \[=S_{\bigtriangleup{AB'M}}\cdot 2 + S_{\bigtriangleup{B'C'M}} + S_{\bigtriangleup{C'DM}}\cdot 2\] \[=2 \cdot \frac{1}{2} \cdot AB' \cdot MH + \frac{1}{2} \cdot B'C' \cdot MH + 2 \cdot \frac{1}{2} \cdot C'D \cdot MH\] \[=2\sqrt{5}+\sqrt{5}+3\sqrt{5}=6\sqrt{5}\]

Finally, the square of the area is $(6\sqrt{5})^2=\boxed{180}$

Solution 2

Denote by $M$ the midpoint of segment $BC$. Let points $P$ and $Q$ be on segment $AD$, such that $AP = AB$ and $DQ = DC$.

Denote $\angle DAM = \alpha$, $\angle BAD = \beta$, $\angle BMA = \theta$, $\angle CMD = \phi$.

Denote $BM = x$. Because $M$ is the midpoint of $BC$, $CM = x$.

Because $AM$ is the angle bisector of $\angle BAD$ and $AB = AP$, $\triangle BAM \cong \triangle PAM$. Hence, $MP = MB$ and $\angle AMP = \theta$. Hence, $\angle MPD = \angle MAP + \angle PMA = \alpha + \theta$.

Because $DM$ is the angle bisector of $\angle CDA$ and $DC = DQ$, $\triangle CDM \cong \triangle QDM$. Hence, $MQ = MC$ and $\angle DMQ = \phi$. Hence, $\angle MQA = \angle MDQ + \angle QMD = \beta + \phi$.

Because $M$ is the midpoint of segment $BC$, $MB = MC$. Because $MP = MB$ and $MQ = MC$, $MP = MQ$.

Thus, $\angle MPD = \angle MQA$.

Thus, \[ \alpha + \theta = \beta + \phi . \hspace{1cm} (1) \]

In $\triangle AMD$, $\angle AMD = 180^\circ - \angle MAD - \angle MDA = 180^\circ - \alpha - \beta$. In addition, $\angle AMD = 180^\circ - \angle BMA - \angle CMD = 180^\circ - \theta - \phi$. Thus, \[ \alpha + \beta = \theta + \phi . \hspace{1cm} (2) \]

Taking $(1) + (2)$, we get $\alpha = \phi$. Taking $(1) - (2)$, we get $\beta = \theta$.

Therefore, $\triangle ADM \sim \triangle AMB \sim \triangle MDC$.

Hence, $\frac{AD}{AM} = \frac{AM}{AB}$ and $\frac{AD}{DM} = \frac{DM}{CD}$. Thus, $AM = \sqrt{AD \cdot AD} = \sqrt{14}$ and $DM = \sqrt{AD \cdot CD} = \sqrt{21}$.

In $\triangle ADM$, by applying the law of cosines, $\cos \angle AMD  = \frac{AM^2 + DM^2 - AD^2}{2 AM \cdot DM} = - \frac{1}{\sqrt{6}}$. Hence, $\sin \angle AMD = \sqrt{1 - \cos^2 \angle AMD} = \frac{\sqrt{5}}{\sqrt{6}}$. Hence, ${\rm Area} \ \triangle ADM = \frac{1}{2} AM \cdot DM \dot \sin \angle AMD = \frac{7 \sqrt{5}}{2}$.

Therefore, \begin{align*} {\rm Area} \ ABCD & = {\rm Area} \ \triangle AMD + {\rm Area} \ \triangle ABM + {\rm Area} \ \triangle MCD \\ & = {\rm Area} \ \triangle AMD \left( 1 + \left( \frac{AM}{AD} \right)^2 + \left( \frac{MD}{AD} \right)^2 \right) \\ & = 6 \sqrt{5} . \end{align*}

Therefore, the square of ${\rm Area} \ ABCD$ is $\left( 6 \sqrt{5} \right)^2 = \boxed{\textbf{(180) }}$.

~Steven Chen (www.professorchenedu.com)

Solution 3 (Visual)

AIME-II-2022-11a.png

Claim

In the triangle $ABC, AB = 2AC, M$ is the midpoint of $AB. D$ is the point of intersection of the circumcircle and the bisector of angle $A.$ Then $DM = BD.$

Proof

Let $A = 2\alpha.$ Then $\angle DBC = \angle DCB = \alpha.$

Let $E$ be the intersection point of the perpendicular dropped from $D$ to $AB$ with the circle.

Then the sum of arcs $\overset{\Large\frown} {BE} + \overset{\Large\frown}{AC} + \overset{\Large\frown}{CD} = 180^\circ.$ \[\overset{\Large\frown} {BE} = 180^\circ – 2\alpha – \overset{\Large\frown}{AC}.\]

Let $E'$ be the point of intersection of the line $CM$ with the circle. $CM$ is perpendicular to $AD, \angle AMC = 90^\circ – \alpha,$ the sum of arcs $\overset{\Large\frown}{A}C +  \overset{\Large\frown}{BE'} = 180^\circ – 2\alpha \implies E'$ coincides with $E.$

The inscribed angles $\angle DEM = \angle DEB, M$ is symmetric to $B$ with respect to $DE, DM = DB.$

Solution

AIME-II-2022-11b.png

Let $AB' = AB, DC' = DC, B'$ and $C'$ on $AD.$

Then $AB' = 2, DC' = 3, B'C' = 2 = AB'.$

Quadrilateral $ABMC'$ is cyclic. Let $\angle A = 2\alpha.$ Then $\angle MBC' = \angle MC'B = \alpha.$

Circle $BB'C'C$ centered at $M, BC$ is its diameter, $\angle BC'C = 90^\circ.$ $\angle DMC' = \angle MC'B,$ since they both complete $\angle MC'C$ to $90^\circ.$

$\angle MB'A = \angle MC'D,$ since they are the exterior angles of an isosceles $\triangle MB'C'.$ $\triangle AMB' \sim \triangle MDC'$ by two angles. $\frac {AB'}{MC'} = \frac {MB'}{DC'}, MC' =\sqrt{AB' \cdot C'D} = \sqrt{6}.$

The height dropped from $M$ to $AD$ is $\sqrt{MB'^2 - (\frac{B'C'}{2})^2} =\sqrt{6 - 1} = \sqrt{5}.$

The areas of triangles $\triangle AMB'$ and $\triangle MC'B'$ are equal to $\sqrt{5},$ area of $\triangle MC'D$ is $\frac{3}{2} \sqrt{5}.$

\[\triangle AMB' = \triangle AMB, \triangle MC'D = \triangle MCD \implies\] The area of $ABCD$ is $(1 + 2 + 3) \sqrt{5} = 6\sqrt{5} \implies 6^2 \cdot 5 = \boxed{180}.$

vladimir.shelomovskii@gmail.com, vvsss

Video Solution by The Power of Logic

https://youtu.be/giLyWHKFr1I

See Also

2022 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
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. AMC logo.png