Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2021 AIME I Problems/Problem 2"

m (→‎Solution 4 (Similar triangles and area): uppercased title.)
(→‎Video Solution)
 
(21 intermediate revisions by 4 users not shown)
Line 23: Line 23:
 
==Solution 1 (Similar Triangles)==
 
==Solution 1 (Similar Triangles)==
 
Let <math>G</math> be the intersection of <math>AD</math> and <math>FC</math>.
 
Let <math>G</math> be the intersection of <math>AD</math> and <math>FC</math>.
−
From vertical angles, we know that <math>\angle FGA= \angle DGC</math>. Also, given that <math>ABCD</math> and <math>AFCE</math> are rectangles, we know that <math>\angle AFG= \angle CDG=90 ^{\circ}</math>.
+
From vertical angles, we know that <math>\angle FGA= \angle DGC</math>. Also, because we are given that <math>ABCD</math> and <math>AFCE</math> are rectangles, we know that <math>\angle AFG= \angle CDG=90 ^{\circ}</math>.
−
Therefore, by AA similarity, we know that triangles <math>AFG</math> and <math>CDG</math> are similar.
+
Therefore, by AA similarity, we know that <math>\triangle AFG\sim\triangle CDG</math>.
  
 
Let <math>AG=x</math>. Then, we have <math>DG=11-x</math>. By similar triangles, we know that <math>FG=\frac{7}{3}(11-x)</math> and <math>CG=\frac{3}{7}x</math>. We have <math>\frac{7}{3}(11-x)+\frac{3}{7}x=FC=9</math>.
 
Let <math>AG=x</math>. Then, we have <math>DG=11-x</math>. By similar triangles, we know that <math>FG=\frac{7}{3}(11-x)</math> and <math>CG=\frac{3}{7}x</math>. We have <math>\frac{7}{3}(11-x)+\frac{3}{7}x=FC=9</math>.
Line 35: Line 35:
 
~yuanyuanC
 
~yuanyuanC
  
−
==Solution 2 (Coordinate Geometry Bash)==
+
== Solution 2 (Similar Triangles) ==
−
Suppose <math>B=(0,0).</math> It follows that
 
−
<cmath>\begin{align*}
 
−
A&=(0,3), \\
 
−
C&=(11,0), \\
 
−
D&=(11,3).
 
−
\end{align*}</cmath>
 
−
 
 
−
Since <math>AECF</math> is a rectangle, we have <math>AE=FC=9</math> and <math>EC=AF=7.</math> The equation of the circle with center <math>A</math> and radius <math>\overline{AE}</math> is <math>x^2+(y-3)^2=81,</math> and the equation of the circle with center <math>C</math> and radius <math>\overline{CE}</math> is <math>(x-11)^2+y^2=49.</math>
 
−
 
 
−
We now have a system of two equations with two variables. Expanding and rearranging respectively give
 
−
<cmath>\begin{align*}
 
−
x^2+y^2-6y&=72, &(1) \\
 
−
x^2+y^2-22x&=-72. &(2)
 
−
\end{align*}</cmath>
 
−
Subtracting <math>(2)</math> from <math>(1),</math> we get <math>22x-6y=144.</math> Simplifying and rearranging produce <cmath>x=\frac{3y+72}{11}. \hspace{34.5mm} (*)</cmath>
 
−
Substituting <math>(*)</math> into <math>(1)</math> gives <cmath>\left(\frac{3y+72}{11}\right)^2+y^2-6y=72,</cmath> which is a quadratic of <math>y.</math> We clear fractions by multiplying both sides by <math>11^2=121,</math> then solve by factoring:
 
−
<cmath>\begin{align*}
 
−
\left(3y+72\right)^2+121y^2-726y&=8712 \\
 
−
\left(9y^2+432y+5184\right)+121y^2-726y&=8712 \\
 
−
130y^2-294y-3528&=0 \\
 
−
2(5x+21)(13x-84)&=0 \\
 
−
y&=-\frac{21}{5},\frac{84}{13}.
 
−
\end{align*}</cmath>
 
−
Since <math>E</math> is in Quadrant IV, we have <math>E=\left(\frac{3\left(-\frac{21}{5}\right)+72}{11},-\frac{21}{5}\right)=\left(\frac{27}{5},-\frac{21}{5}\right).</math> It follows that the equation of <math>\overleftrightarrow{AE}</math> is <math>y=-\frac{4}{3}x+3.</math>
 
  
−
Let <math>G</math> be the intersection of <math>\overline{AD}</math> and <math>\overline{FC},</math> and <math>H</math> be the intersection of <math>\overline{AE}</math> and <math>\overline{BC}.</math> Since <math>H</math> is the <math>x</math>-intercept of <math>\overleftrightarrow{AE},</math> we obtain <math>H=\left(\frac94,0\right).</math>
+
Again, let the intersection of <math>AE</math> and <math>BC</math> be <math>G</math>. By AA similarity, <math>\triangle AFG \sim \triangle CDG</math> with a <math>\frac{7}{3}</math> ratio. Define <math>x</math> as <math>\frac{[CDG]}{9}</math>. Because of similar triangles, <math>[AFG] = 49x</math>. Using <math>ABCD</math>, the area of the parallelogram is <math>33-18x</math>. Using <math>AECF</math>, the area of the parallelogram is <math>63-98x</math>. These equations are equal, so we can solve for <math>x</math> and obtain <math>x = \frac{3}{8}</math>. Thus, <math>18x = \frac{27}{4}</math>, so the area of the parallelogram is <math>33 - \frac{27}{4} = \frac{105}{4}</math>.
  
−
By symmetry, quadrilateral <math>AGCH</math> is a parallelogram. Its area is <math>HC\cdot AB=\left(11-\frac94\right)\cdot3=\frac{105}{4},</math> from which the requested sum is <math>105+4=\boxed{109}.</math>  
+
Finally, the answer is <math>105+4=\boxed{109}</math>.
  
−
~MRENTHUSIASM
+
~mathboy100
  
 
==Solution 3 (Pythagorean Theorem)==
 
==Solution 3 (Pythagorean Theorem)==
Line 75: Line 51:
 
By the Pythagorean theorem again, <math>{CG}^2={EG}^2+{CE}^2</math>: <cmath>11-x=\sqrt{7^2+(9-\sqrt{x^2+9})^2}.</cmath>
 
By the Pythagorean theorem again, <math>{CG}^2={EG}^2+{CE}^2</math>: <cmath>11-x=\sqrt{7^2+(9-\sqrt{x^2+9})^2}.</cmath>
  
−
Solving, we get <math>x=\frac{9}{4}</math>, so the area of the parallelogram is <math>3\cdot(11-\frac{9}{4})=\frac{105}{4}</math>, and <math>105+4=\framebox{109}</math>.
+
Solving, we get <math>x=\frac{9}{4}</math>, so the area of the parallelogram is <math>3\cdot\left(11-\frac{9}{4}\right)=\frac{105}{4}</math>, and <math>105+4=\framebox{109}</math>.
  
 
~JulianaL25
 
~JulianaL25
  
−
== Solution 4 (Similar Triangles and Area) ==
+
== Solution 4 (Pythagorean Theorem)==
−
 
 
−
Again, let the intersection of <math>AE</math> and <math>BC</math> be <math>G</math>. By AA similarity, <math>AFG \sim CDG</math> with a <math>\frac{7}{3}</math> ratio. Define <math>x</math> as <math>\frac{[CDG]}{9}</math>. Because of similar triangles, <math>[AFG] = 49x</math>. Using <math>ABCD</math>, the area of the parallelogram is <math>33-18x</math>. Using <math>AECF</math>, the area of the parallelogram is <math>63-98x</math>. These equations are equal, so we can solve for <math>x</math> and obtain <math>x = \frac{3}{8}</math>. Thus, <math>18x = \frac{27}{4}</math>, so the area of the parallelogram is <math>33 - \frac{27}{4} = \frac{105}{4}</math>.
 
−
 
 
−
~mathboy100
 
−
 
 
−
== Solution 5 ==
 
  
 
Let <math>P = AD \cap FC</math>, and <math>K = AE \cap BC</math>. Also let <math>AP = x</math>.
 
Let <math>P = AD \cap FC</math>, and <math>K = AE \cap BC</math>. Also let <math>AP = x</math>.
Line 110: Line 80:
 
This simplifies (not-as-shortly) to <math>x = \dfrac{35}{4}</math>. Now we have to solve for the area of <math>APCK</math>. We know that the height is <math>3</math> because the height of the parallelogram is the same as the height of the smaller rectangle.
 
This simplifies (not-as-shortly) to <math>x = \dfrac{35}{4}</math>. Now we have to solve for the area of <math>APCK</math>. We know that the height is <math>3</math> because the height of the parallelogram is the same as the height of the smaller rectangle.
  
−
From the area of a parallelogram (we know that the base is <math>\dfrac{35}{4}</math> and the height is <math>3</math>), it is clear that the area is <math>\dfrac{105}{4}</math>, giving an answer of <math>\boxed{109}</math>. ~ishanvannadil2008 for the solution sketch, Tuatara for rephrasing and <math>\LaTeX</math>.
+
From the area of a parallelogram (we know that the base is <math>\dfrac{35}{4}</math> and the height is <math>3</math>), it is clear that the area is <math>\dfrac{105}{4}</math>, giving an answer of <math>\boxed{109}</math>.  
  
−
==Solution 6 (Trigbash)==
+
~ishanvannadil2008 (Solution Sketch)
 +
 
 +
~Tuatara (Rephrasing and <math>\LaTeX</math>)
 +
 
 +
==Solution 5 (Coordinate Geometry)==
 +
Suppose <math>B=(0,0).</math> It follows that <math>A=(0,3),C=(11,0),</math> and <math>D=(11,3).</math>
 +
 
 +
Since <math>AECF</math> is a rectangle, we have <math>AE=FC=9</math> and <math>EC=AF=7.</math> The equation of the circle with center <math>A</math> and radius <math>\overline{AE}</math> is <math>x^2+(y-3)^2=81,</math> and the equation of the circle with center <math>C</math> and radius <math>\overline{CE}</math> is <math>(x-11)^2+y^2=49.</math>
 +
 
 +
We now have a system of two equations with two variables. Expanding and rearranging respectively give
 +
<cmath>\begin{align*}
 +
x^2+y^2-6y&=72, &(1) \\
 +
x^2+y^2-22x&=-72. &(2)
 +
\end{align*}</cmath>
 +
Subtracting <math>(2)</math> from <math>(1),</math> we obtain <math>22x-6y=144.</math> Simplifying and rearranging produce <cmath>x=\frac{3y+72}{11}. \hspace{34.5mm} (*)</cmath>
 +
Substituting <math>(*)</math> into <math>(1)</math> gives <cmath>\left(\frac{3y+72}{11}\right)^2+y^2-6y=72,</cmath> which is a quadratic of <math>y.</math> We clear fractions by multiplying both sides by <math>11^2=121,</math> then solve by factoring:
 +
<cmath>\begin{align*}
 +
\left(3y+72\right)^2+121y^2-726y&=8712 \\
 +
\left(9y^2+432y+5184\right)+121y^2-726y&=8712 \\
 +
130y^2-294y-3528&=0 \\
 +
2(5y+21)(13y-84)&=0 \\
 +
y&=-\frac{21}{5},\frac{84}{13}.
 +
\end{align*}</cmath>
 +
Since <math>E</math> is in Quadrant IV, we have <math>E=\left(\frac{3\left(-\frac{21}{5}\right)+72}{11},-\frac{21}{5}\right)=\left(\frac{27}{5},-\frac{21}{5}\right).</math> It follows that the equation of <math>\overleftrightarrow{AE}</math> is <math>y=-\frac{4}{3}x+3.</math>
 +
 
 +
Let <math>G</math> be the intersection of <math>\overline{AD}</math> and <math>\overline{FC},</math> and <math>H</math> be the intersection of <math>\overline{AE}</math> and <math>\overline{BC}.</math> Since <math>H</math> is the <math>x</math>-intercept of <math>\overleftrightarrow{AE},</math> we get <math>H=\left(\frac94,0\right).</math>
 +
 
 +
By symmetry, quadrilateral <math>AGCH</math> is a parallelogram. Its area is <math>HC\cdot AB=\left(11-\frac94\right)\cdot3=\frac{105}{4},</math> from which the requested sum is <math>105+4=\boxed{109}.</math>
 +
 
 +
~MRENTHUSIASM
 +
 
 +
==Solution 6 (Trigonometry)==
  
 
Let the intersection of <math>AE</math> and <math>BC</math> be <math>G</math>. It is useful to find <math>\tan(\angle DAE)</math>, because  <math>\tan(\angle DAE)=\frac{3}{BG}</math>  and  <math>\frac{3}{\tan(\angle DAE)}=BG</math>. From there, subtracting the areas of the two triangles from the larger rectangle, we get  Area = <math>33-3BG=33-\frac{9}{\tan(\angle DAE)}</math>.
 
Let the intersection of <math>AE</math> and <math>BC</math> be <math>G</math>. It is useful to find <math>\tan(\angle DAE)</math>, because  <math>\tan(\angle DAE)=\frac{3}{BG}</math>  and  <math>\frac{3}{\tan(\angle DAE)}=BG</math>. From there, subtracting the areas of the two triangles from the larger rectangle, we get  Area = <math>33-3BG=33-\frac{9}{\tan(\angle DAE)}</math>.
Line 124: Line 125:
 
<math>\tan(\angle DAE) = \tan\left(\tan^{-1}\left(\frac{3}{11}\right)+\tan^{-1}\left(\frac{7}{9}\right)\right) = \frac{\frac{3}{11}+\frac{7}{9}}{1-\frac{3}{11}\cdot\frac{7}{9}} = \frac{\frac{104}{99}}{\frac{78}{99}} = \frac{4}{3}</math>  
 
<math>\tan(\angle DAE) = \tan\left(\tan^{-1}\left(\frac{3}{11}\right)+\tan^{-1}\left(\frac{7}{9}\right)\right) = \frac{\frac{3}{11}+\frac{7}{9}}{1-\frac{3}{11}\cdot\frac{7}{9}} = \frac{\frac{104}{99}}{\frac{78}{99}} = \frac{4}{3}</math>  
  
−
Area =     <math>33-\frac{9}{\frac{4}{3}} = 33-\frac{27}{4 } = \frac{105}{4}</math>. The answer is <math>105+4=109</math>.
+
<math>\mathrm{Area}=33-\frac{9}{\frac{4}{3}} = 33-\frac{27}{4 } = \frac{105}{4}</math>. The answer is <math>105+4=\boxed{109}</math>.
−
~ twotothetenthis1024
+
 
 +
~twotothetenthis1024
  
 
==Video Solution by Punxsutawney Phil==
 
==Video Solution by Punxsutawney Phil==
Line 138: Line 140:
 
==Video Solution==
 
==Video Solution==
 
https://www.youtube.com/watch?v=BinfKrc5bWo
 
https://www.youtube.com/watch?v=BinfKrc5bWo
 +
 +
==Video Solution by Power of Logic==
 +
https://youtu.be/WS6X1MQ37jg
  
 
==See Also==
 
==See Also==

Latest revision as of 23:59, 22 August 2022

Problem

In the diagram below, $ABCD$ is a rectangle with side lengths $AB=3$ and $BC=11$, and $AECF$ is a rectangle with side lengths $AF=7$ and $FC=9,$ as shown. The area of the shaded region common to the interiors of both rectangles is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] pair A, B, C, D, E, F; A=(0,3); B=(0,0); C=(11,0); D=(11,3); E=foot(C, A, (9/4,0)); F=foot(A, C, (35/4,3)); draw(A--B--C--D--cycle); draw(A--E--C--F--cycle); filldraw(A--(9/4,0)--C--(35/4,3)--cycle,gray*0.5+0.5*lightgray); dot(A^^B^^C^^D^^E^^F); label("$A$", A, W); label("$B$", B, W); label("$C$", C, (1,0)); label("$D$", D, (1,0)); label("$F$", F, N); label("$E$", E, S); [/asy]

Solution 1 (Similar Triangles)

Let $G$ be the intersection of $AD$ and $FC$. From vertical angles, we know that $\angle FGA= \angle DGC$. Also, because we are given that $ABCD$ and $AFCE$ are rectangles, we know that $\angle AFG= \angle CDG=90 ^{\circ}$. Therefore, by AA similarity, we know that $\triangle AFG\sim\triangle CDG$.

Let $AG=x$. Then, we have $DG=11-x$. By similar triangles, we know that $FG=\frac{7}{3}(11-x)$ and $CG=\frac{3}{7}x$. We have $\frac{7}{3}(11-x)+\frac{3}{7}x=FC=9$.

Solving for $x$, we have $x=\frac{35}{4}$. The area of the shaded region is just $3\cdot \frac{35}{4}=\frac{105}{4}$.

Thus, the answer is $105+4=\framebox{109}$.

~yuanyuanC

Solution 2 (Similar Triangles)

Again, let the intersection of $AE$ and $BC$ be $G$. By AA similarity, $\triangle AFG \sim \triangle CDG$ with a $\frac{7}{3}$ ratio. Define $x$ as $\frac{[CDG]}{9}$. Because of similar triangles, $[AFG] = 49x$. Using $ABCD$, the area of the parallelogram is $33-18x$. Using $AECF$, the area of the parallelogram is $63-98x$. These equations are equal, so we can solve for $x$ and obtain $x = \frac{3}{8}$. Thus, $18x = \frac{27}{4}$, so the area of the parallelogram is $33 - \frac{27}{4} = \frac{105}{4}$.

Finally, the answer is $105+4=\boxed{109}$.

~mathboy100

Solution 3 (Pythagorean Theorem)

Let the intersection of $AE$ and $BC$ be $G$, and let $BG=x$, so $CG=11-x$.

By the Pythagorean theorem, ${AG}^2={AB}^2+{BG}^2$, so $AG=\sqrt{x^2+9}$, and thus $EG=9-\sqrt{x^2+9}$.

By the Pythagorean theorem again, ${CG}^2={EG}^2+{CE}^2$: \[11-x=\sqrt{7^2+(9-\sqrt{x^2+9})^2}.\]

Solving, we get $x=\frac{9}{4}$, so the area of the parallelogram is $3\cdot\left(11-\frac{9}{4}\right)=\frac{105}{4}$, and $105+4=\framebox{109}$.

~JulianaL25

Solution 4 (Pythagorean Theorem)

Let $P = AD \cap FC$, and $K = AE \cap BC$. Also let $AP = x$.

$CK$ also has to be $x$ by parallelogram properties. Then $PD$ and $BK$ must be $11-x$ because the sum of the segments has to be $11$.

We can easily solve for $PC$ by the Pythagorean Theorem: \begin{align*} DC^2 + PD^2 &= PC^2\\ 9 + (11-x)^2 &= PC^2 \end{align*} It follows shortly that $PC = \sqrt{x^2-22x+30}$.

Also, $FC = 9$, and $FP + PC = 9$. We can then say that $PC = \sqrt{x^2-22x+30}$, so $FP = 9 - \sqrt{x^2-22x+30}$.

Now we can apply the Pythagorean Theorem to $\triangle AFP$. \begin{align*} AF^2 + FP^2 = AP^2\\ 49 + \left(9 - \sqrt{x^2-22x+30}\right)^2 = x^2 \end{align*}

This simplifies (not-as-shortly) to $x = \dfrac{35}{4}$. Now we have to solve for the area of $APCK$. We know that the height is $3$ because the height of the parallelogram is the same as the height of the smaller rectangle.

From the area of a parallelogram (we know that the base is $\dfrac{35}{4}$ and the height is $3$), it is clear that the area is $\dfrac{105}{4}$, giving an answer of $\boxed{109}$.

~ishanvannadil2008 (Solution Sketch)

~Tuatara (Rephrasing and $\LaTeX$)

Solution 5 (Coordinate Geometry)

Suppose $B=(0,0).$ It follows that $A=(0,3),C=(11,0),$ and $D=(11,3).$

Since $AECF$ is a rectangle, we have $AE=FC=9$ and $EC=AF=7.$ The equation of the circle with center $A$ and radius $\overline{AE}$ is $x^2+(y-3)^2=81,$ and the equation of the circle with center $C$ and radius $\overline{CE}$ is $(x-11)^2+y^2=49.$

We now have a system of two equations with two variables. Expanding and rearranging respectively give \begin{align*} x^2+y^2-6y&=72, &(1) \\ x^2+y^2-22x&=-72. &(2) \end{align*} Subtracting $(2)$ from $(1),$ we obtain $22x-6y=144.$ Simplifying and rearranging produce \[x=\frac{3y+72}{11}. \hspace{34.5mm} (*)\] Substituting $(*)$ into $(1)$ gives \[\left(\frac{3y+72}{11}\right)^2+y^2-6y=72,\] which is a quadratic of $y.$ We clear fractions by multiplying both sides by $11^2=121,$ then solve by factoring: \begin{align*} \left(3y+72\right)^2+121y^2-726y&=8712 \\ \left(9y^2+432y+5184\right)+121y^2-726y&=8712 \\ 130y^2-294y-3528&=0 \\ 2(5y+21)(13y-84)&=0 \\ y&=-\frac{21}{5},\frac{84}{13}. \end{align*} Since $E$ is in Quadrant IV, we have $E=\left(\frac{3\left(-\frac{21}{5}\right)+72}{11},-\frac{21}{5}\right)=\left(\frac{27}{5},-\frac{21}{5}\right).$ It follows that the equation of $\overleftrightarrow{AE}$ is $y=-\frac{4}{3}x+3.$

Let $G$ be the intersection of $\overline{AD}$ and $\overline{FC},$ and $H$ be the intersection of $\overline{AE}$ and $\overline{BC}.$ Since $H$ is the $x$-intercept of $\overleftrightarrow{AE},$ we get $H=\left(\frac94,0\right).$

By symmetry, quadrilateral $AGCH$ is a parallelogram. Its area is $HC\cdot AB=\left(11-\frac94\right)\cdot3=\frac{105}{4},$ from which the requested sum is $105+4=\boxed{109}.$

~MRENTHUSIASM

Solution 6 (Trigonometry)

Let the intersection of $AE$ and $BC$ be $G$. It is useful to find $\tan(\angle DAE)$, because  $\tan(\angle DAE)=\frac{3}{BG}$  and  $\frac{3}{\tan(\angle DAE)}=BG$. From there, subtracting the areas of the two triangles from the larger rectangle, we get  Area = $33-3BG=33-\frac{9}{\tan(\angle DAE)}$.

let $\angle CAD = \alpha$. Let $\angle CAE = \beta$. Note, $\alpha+\beta=\angle DAE$.

$\alpha=\tan^{-1}\left(\frac{3}{11}\right)$

$\beta=\tan^{-1}\left(\frac{7}{9}\right)$

$\tan(\angle DAE) = \tan\left(\tan^{-1}\left(\frac{3}{11}\right)+\tan^{-1}\left(\frac{7}{9}\right)\right) = \frac{\frac{3}{11}+\frac{7}{9}}{1-\frac{3}{11}\cdot\frac{7}{9}} = \frac{\frac{104}{99}}{\frac{78}{99}} = \frac{4}{3}$  

$\mathrm{Area}=33-\frac{9}{\frac{4}{3}} = 33-\frac{27}{4 } = \frac{105}{4}$. The answer is $105+4=\boxed{109}$.

~twotothetenthis1024

Video Solution by Punxsutawney Phil

https://youtube.com/watch?v=H17E9n2nIyY&t=289s

Video Solution

https://youtu.be/M3DsERqhiDk?t=275

Video Solution by Steven Chen (in Chinese)

https://youtu.be/eaS5gRLSqgY

Video Solution

https://www.youtube.com/watch?v=BinfKrc5bWo

Video Solution by Power of Logic

https://youtu.be/WS6X1MQ37jg

See Also

2021 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2021_AIME_I_Problems/Problem_2&oldid=177374"