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Difference between revisions of "2021 AIME II Problems/Problem 12"

(→‎Solution 3 (Pythagorean Theorem and Right Triangle Trigonometry))
(→‎Solution 3 (Pythagorean Theorem and Right Triangle Trigonometry))
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Let <math>BC'=p,C'E=q,EA'=r,A'D=s,AA'=h_1,</math> and <math>CC'=h_2.</math> Applying the Pythagorean Theorem to right triangles <math>\triangle ABA',\triangle BCC',\triangle CDC',</math> and <math>\triangle DAA',</math> we respectively get
 
Let <math>BC'=p,C'E=q,EA'=r,A'D=s,AA'=h_1,</math> and <math>CC'=h_2.</math> Applying the Pythagorean Theorem to right triangles <math>\triangle ABA',\triangle BCC',\triangle CDC',</math> and <math>\triangle DAA',</math> we respectively get
 
<cmath>\begin{array}{ccccccccccccccccc}
 
<cmath>\begin{array}{ccccccccccccccccc}
−
(p+q+r)^2&+&h_1^2&=&5^2, &&&&&&&&&&&&(1) \\ [1ex]
+
(p+q+r)^2&+&h_1^2&=&5^2, &&&&&&&&&&&&\hspace{36mm}(1) \\ [1ex]
−
p^2&+&h_2^2&=&6^2, &&&&&&&&&&&&(2) \\ [1ex]
+
p^2&+&h_2^2&=&6^2, &&&&&&&&&&&&\hspace{36mm}(2) \\ [1ex]
−
(q+r+s)^2&+&h_2^2&=&9^2, &&&&&&&&&&&&(3) \\ [1ex]
+
(q+r+s)^2&+&h_2^2&=&9^2, &&&&&&&&&&&&\hspace{36mm}(3) \\ [1ex]
−
s^2&+&h_2^2&=&7^2. &&&&&&&&&&&&(4)
+
s^2&+&h_2^2&=&7^2. &&&&&&&&&&&&\hspace{36mm}(4)
 
\end{array}</cmath>
 
\end{array}</cmath>
 
Let the brackets denote areas. We get
 
Let the brackets denote areas. We get
Line 220: Line 220:
 
\frac12(p+q+r+s)h_1+\frac12(p+q+r+s)h_2&=30 \\
 
\frac12(p+q+r+s)h_1+\frac12(p+q+r+s)h_2&=30 \\
 
\frac12(p+q+r+s)(h_1+h_2)&=30 \\
 
\frac12(p+q+r+s)(h_1+h_2)&=30 \\
−
(p+q+r+s)(h_1+h_2)&=60. \hspace{13.25mm}(5)
+
(p+q+r+s)(h_1+h_2)&=60. \hspace{49.25mm}(5)
 
\end{align*}</cmath>
 
\end{align*}</cmath>
 
We subtract <math>(2)+(4)</math> from <math>(1)+(3):</math>
 
We subtract <math>(2)+(4)</math> from <math>(1)+(3):</math>
Line 227: Line 227:
 
\left[(p+q+r)^2-s^2\right]+\left[(q+r+s)^2-p^2\right]&=21 \\
 
\left[(p+q+r)^2-s^2\right]+\left[(q+r+s)^2-p^2\right]&=21 \\
 
(p+q+r+s)(p+q+r-s)+(p+q+r+s)(-p+q+r+s)&=21 \\
 
(p+q+r+s)(p+q+r-s)+(p+q+r+s)(-p+q+r+s)&=21 \\
−
(p+q+r+s)(q+r)&=21.
+
(p+q+r+s)(q+r)&=21. \hspace{10mm}(6)
 
\end{align*}</cmath>
 
\end{align*}</cmath>
 +
From right triangles <math>\triangle AEA'</math> and <math>\triangle CEC',</math> we have <math>\tan\theta=\frac{h_1}{r}=\frac{h_2}{q}.</math> It follows that
 +
<cmath>\begin{alignat*}{8}
 +
\tan\theta&=\frac{h_1}{r}\qquad&\implies\qquad h_1&=r\tan\theta, \hspace{64mm}&(1\star)\\
 +
\tan\theta&=\frac{h_2}{q}\qquad&\implies\qquad h_2&=q\tan\theta. &(2\star)
 +
\end{alignat*}</cmath>
 +
Finally, dividing <math>(5)</math> by <math>(6)</math> gives
  
 
<b>NO EDIT PLEASE. WILL BE BACK SOON.</b>
 
<b>NO EDIT PLEASE. WILL BE BACK SOON.</b>

Revision as of 00:17, 1 June 2021

Problem

A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\tan \theta$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Diagram

File:2021 AIME II Problem 12 Diagram.png

~MRENTHUSIASM (by Geometry Expressions)

Solution 1

We denote by $A$, $B$, $C$ and $D$ four vertices of this quadrilateral, such that $AB = 5$, $BC = 6$, $CD = 9$, $DA = 7$. We denote by $E$ the point that two diagonals $AC$ and $BD$ meet at. To simplify the notation, we denote $a = AE$, $b = BE$, $c = CE$, $d = DE$.

We denote $\theta = \angle AED$. Hence, $\angle AEB = \angle CED = 180^\circ - \theta$ and $\angle BEC = \theta$.

First, we use the triangle area formula with sines to write down an equation of the area of the quadrilateral $ABCD$.

We have \begin{align*} {\rm Area} \ ABCD & = {\rm Area} \ \triangle ABE + {\rm Area} \ \triangle BCE + {\rm Area} \ \triangle CDE + {\rm Area} \ \triangle DAE \\ & = \frac{1}{2} ab \sin \angle AEB + \frac{1}{2} bc \sin \angle BEC + \frac{1}{2} cd \sin \angle CED + \frac{1}{2} da \sin \angle DEA \\ & = \frac{1}{2} ab \sin \theta + \frac{1}{2} bc \sin \left( 180^\circ - \theta \right) + \frac{1}{2} cd \sin \theta + \frac{1}{2} da \sin \left( 180^\circ - \theta \right) \\ & = \frac{1}{2} ab \sin \theta + \frac{1}{2} bc \sin \theta + \frac{1}{2} cd \sin \theta + \frac{1}{2} da \sin \theta \\ & = \frac{1}{2} \left( ab + bc + cd + da \right) \sin \theta , \end{align*} where the second equality follows from the formula to use the sine function to compute a triangle area, the the fourth equality follows from the property that $\sin \left( 180^\circ - \theta \right) = \sin \theta$.

Because ${\rm Area} \ ABCD = 30$, we have \[ \left( ab + bc + cd + da \right) \sin \theta = 60 . \ \ \ (1) \].

Second, we use the law of cosines to establish four equations for four sides of the quadrilateral $ABCD$.

In $\triangle AEB$, following from the law of cosines, we have \[ a^2 + b^2 - 2 a b \cos \angle AEB = AB^2 . \]

Because $\cos \angle AEB = \cos \left( 180^\circ - \theta \right) = \cos \theta$ and $AB = 5$, we have \[ a^2 + b^2 + 2 a b \cos \theta = 5^2 . \ \ \ (2) \]

In $\triangle BEC$, following from the law of cosines, we have \[ b^2 + c^2 - 2 b c \cos \angle BEC = BC^2 . \]

Because $\cos \angle AEB = \cos \theta$ and $BC = 6$, we have \[ b^2 + c^2 - 2 b c \cos \theta = 6^2 . \ \ \ (3) \]

In $\triangle CED$, following from the law of cosines, we have \[ c^2 + d^2 - 2 c d \cos \angle CED = CD^2 . \]

Because $\cos \angle CED = \cos \left( 180^\circ - \theta \right) = \cos \theta$ and $CD = 9$, we have \[ c^2 + d^2 + 2 c d \cos \theta = 9^2 . \ \ \ (4) \]

In $\triangle DEA$, following from the law of cosines, we have \[ d^2 + a^2 - 2 d a \cos \angle DEA = DA^2 . \]

Because $\cos \angle DEC = \cos \theta$ and $DA = 7$, we have \[ d^2 + a^2 - 2 d a \cos \theta = 7^2 . \ \ \ (5) \]

By taking $\frac{1}{2} \left( {\rm Eq} \ (2) - {\rm Eq} \ (3) + {\rm Eq} \ (4) - {\rm Eq} \ (5) \right)$, we get

\[ \left( ab + bc + cd + da \right) \cos \theta = \frac{21}{2} . \ \ \ (6) \]

By taking $\frac{{\rm Eq} \ (1)}{{\rm Eq} \ (6)}$, we get \[ \tan \theta = \frac{60}{21/2} = \frac{40}{7} . \]

Therefore, by writing this answer in the form of $\frac{m}{n}$, we have $m = 40$ and $n = 7$. Therefore, the answer to this question is $m + n = 40 + 7 = \boxed{047}$.

~ Steven Chen (www.professorchenedu.com)

Solution 2

Since we are asked to find $\tan \theta$, we can find $\sin \theta$ and $\cos \theta$ separately and use their values to get $\tan \theta$. We can start by drawing a diagram. Let the vertices of the quadrilateral be $A$, $B$, $C$, and $D$. Let $AB = 5$, $BC = 6$, $CD = 9$, and $DA = 7$. Let $AX = a$, $BX = b$, $CX = c$, and $DX = d$. We know that $\theta$ is the acute angle formed between the intersection of the diagonals $AC$ and $BD$.

[asy] unitsize(4cm); pair A,B,C,D,X; A = (0,0); B = (1,0); C = (1.25,-1); D = (-0.75,-0.75); draw(A--B--C--D--cycle,black+1bp); X = intersectionpoint(A--C,B--D); draw(A--C); draw(B--D); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); dot(X); label("$X$",X,S); label("$5$",(A+B)/2,N); label("$6$",(B+C)/2,E); label("$9$",(C+D)/2,S); label("$7$",(D+A)/2,W); label("$\theta$",X,2.5E); label("$a$",(A+X)/2,NE); label("$b$",(B+X)/2,NW); label("$c$",(C+X)/2,SW); label("$d$",(D+X)/2,SE); [/asy]

We are given that the area of quadrilateral $ABCD$ is $30$. We can express this area using the areas of triangles $AXB$, $BXC$, $CXD$, and $DXA$. Since we want to find $\sin \theta$ and $\cos \theta$, we can represent these areas using $\sin \theta$ as follows:

\begin{align*} 30 &=[ABCD] \\ &=[AXB] + [BXC] + [CXD] + [DXA] \\ &=\frac{1}{2} ab \sin (\angle AXB) + \frac{1}{2} bc \sin (\angle BXC) + \frac{1}{2} cd \sin (\angle CXD) + \frac{1}{2} da \sin (\angle AXD) \\ &=\frac{1}{2} ab \sin (180^\circ - \theta) + \frac{1}{2} bc \sin (\theta) + \frac{1}{2} cd \sin (180^\circ - \theta) + \frac{1}{2} da \sin (\theta) \end{align*}

We know that $\sin (180^\circ - \theta) = \sin \theta$. Therefore it follows that:

\begin{align*} 30 &=\frac{1}{2} ab \sin (180^\circ - \theta) + \frac{1}{2} bc \sin (\theta) + \frac{1}{2} cd \sin (180^\circ - \theta) + \frac{1}{2} da \sin (\theta) \\ &=\frac{1}{2} ab \sin (\theta) + \frac{1}{2} bc \sin (\theta) + \frac{1}{2} cd \sin (\theta) + \frac{1}{2} da \sin (\theta) \\ &=\frac{1}{2}\sin\theta (ab + bc + cd + da) \end{align*}

From here we see that $\sin \theta = \frac{60}{ab + bc + cd + da}$. Now we need to find $\cos \theta$. Using the Law of Cosines on each of the four smaller triangles, we get following equations:

\begin{align*} 5^2 &= a^2 + b^2 - 2ab\cos(180^\circ-\theta) \\ 6^2 &= b^2 + c^2 - 2bc\cos \theta \\ 9^2 &= c^2 + d^2 - 2cd\cos(180^\circ-\theta) \\ 7^2 &= d^2 + a^2 - 2da\cos \theta \end{align*}

We know that $\cos (180^\circ - \theta) = -\cos \theta$. We can substitute this value into our equations to get:

\begin{align*} 5^2 &= a^2 + b^2 + 2ab\cos \theta \\ 6^2 &= b^2 + c^2 - 2bc\cos \theta \\ 9^2 &= c^2 + d^2 + 2cd\cos \theta \\ 7^2 &= d^2 + a^2 - 2da\cos \theta \end{align*}

If we subtract the sum of the first and third equation from the sum of the second and fourth equation, the squared terms cancel, leaving us with: \[5^2 + 9^2 - 6^2 - 7^2 = 2ab \cos \theta + 2bc \cos \theta + 2cd \cos \theta + 2da \cos \theta\] \[21 = 2\cos \theta (ab + bc + cd + da)\]

From here we see that $\cos \theta = \frac{21/2}{ab + bc + cd + da}$.

Since we have figured out $\sin \theta$ and $\cos \theta$, we can calculate $\tan \theta$:

\[\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{60}{ab + bc + cd + da}}{\frac{21/2}{ab + bc + cd + da}} = \frac{60}{21/2} = \frac{120}{21} = \frac{40}{7}\]

Therefore our answer is $40 + 7 = \boxed{047}$.

~ my_aops_lessons

Solution 3 (Pythagorean Theorem and Right Triangle Trigonometry)

This solution refers to the Diagram section.

In convex quadrilateral $ABCD,$ let $AB=5,BC=6,CD=9,$ and $DA=7.$ Let $A'$ and $C'$ be the feet of the perpendiculars from $A$ and $C,$ respectively, to $\overline{BD}.$ We obtain the following diagram:

File:2021 AIME II Problem 12 Solution.png

Let $BC'=p,C'E=q,EA'=r,A'D=s,AA'=h_1,$ and $CC'=h_2.$ Applying the Pythagorean Theorem to right triangles $\triangle ABA',\triangle BCC',\triangle CDC',$ and $\triangle DAA',$ we respectively get \[\begin{array}{ccccccccccccccccc} (p+q+r)^2&+&h_1^2&=&5^2, &&&&&&&&&&&&\hspace{36mm}(1) \\ [1ex] p^2&+&h_2^2&=&6^2, &&&&&&&&&&&&\hspace{36mm}(2) \\ [1ex] (q+r+s)^2&+&h_2^2&=&9^2, &&&&&&&&&&&&\hspace{36mm}(3) \\ [1ex] s^2&+&h_2^2&=&7^2. &&&&&&&&&&&&\hspace{36mm}(4) \end{array}\] Let the brackets denote areas. We get \begin{align*} [ABD]+[CBD]&=[ABCD] \\ \frac12(p+q+r+s)h_1+\frac12(p+q+r+s)h_2&=30 \\ \frac12(p+q+r+s)(h_1+h_2)&=30 \\ (p+q+r+s)(h_1+h_2)&=60. \hspace{49.25mm}(5) \end{align*} We subtract $(2)+(4)$ from $(1)+(3):$ \begin{align*} (p+q+r)^2+(q+r+s)^2-p^2-s^2&=21 \\ \left[(p+q+r)^2-s^2\right]+\left[(q+r+s)^2-p^2\right]&=21 \\ (p+q+r+s)(p+q+r-s)+(p+q+r+s)(-p+q+r+s)&=21 \\ (p+q+r+s)(q+r)&=21. \hspace{10mm}(6) \end{align*} From right triangles $\triangle AEA'$ and $\triangle CEC',$ we have $\tan\theta=\frac{h_1}{r}=\frac{h_2}{q}.$ It follows that \begin{alignat*}{8} \tan\theta&=\frac{h_1}{r}\qquad&\implies\qquad h_1&=r\tan\theta, \hspace{64mm}&(1\star)\\ \tan\theta&=\frac{h_2}{q}\qquad&\implies\qquad h_2&=q\tan\theta. &(2\star) \end{alignat*} Finally, dividing $(5)$ by $(6)$ gives

NO EDIT PLEASE. WILL BE BACK SOON.

~MRENTHUSIASM

Video Solution

https://www.youtube.com/watch?v=7DxIdTLNbo0

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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