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Difference between revisions of "2018 AIME II Problems/Problem 12"

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label("$\Lambda$", D--P/2, 3*NW);
 
label("$\Lambda$", D--P/2, 3*NW);
 
</asy>
 
</asy>
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==Solution 1==
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Let <math>AP=x</math> and let <math>PC=\rho x</math>. Let <math>[ABP]=\Delta</math> and let <math>[ADP]=\Lambda</math>. We easily get <math>[PBC]=\rho \Delta</math> and <math>[PCD]=\rho\Lambda</math>.
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We are given that <math>[APB] +[CPD] = [BPC]+[APD]</math>, which we can now write as <cmath>\Delta + \rho\Lambda = \rho\Delta + \Lambda \qquad \Longrightarrow \qquad \Delta -\Lambda = \rho (\Delta -\Lambda).</cmath> Either <math>\Delta = \Lambda</math> or <math>\rho=1</math>. The former would imply that <math>ABCD</math> is a parallelogram, which it isn't; therefore we conclude <math>\rho=1</math> and <math>P</math> is the midpoint of <math>AC</math>. Let <math>\angle BAD = \theta</math> and <math>\angle BCD = \phi</math>. Then <math>[ABCD]=2\cdot [BCD]=140\sin\phi</math>. On one hand, since <math>[ABD]=[BCD]</math>, we have <cmath>\begin{align}\sqrt{65}\sin\theta = 7\sin\phi \quad \implies \quad 16+49\cos^2\phi = 65\cos^2\theta\end{align}</cmath>whereas, on the other hand, using cosine formula to get the length of <math>BD</math>, we get <cmath>10^2+4\cdot 65 - 40\sqrt{65}\cos\theta = 10^2+14^2-280\cos\phi</cmath><cmath>\begin{align}\tag{2}\implies \qquad  65\cos^2\theta = \left(7\cos\phi+ \frac{8}{5}\right)^2\end{align}</cmath>Eliminating <math>\cos\theta</math> in the above two equations and solving for <math>\cos\phi</math> we get<cmath>\cos\phi = -\frac{3}{5}\qquad \implies \qquad \sin\phi = \frac{4}{5}</cmath>which finally yields <math>[ABCD]=2\cdot [BCD] = 140\sin\phi = 112</math>.
  
 
==Solution 1==
 
==Solution 1==

Revision as of 15:22, 29 November 2021

Problem

Let $ABCD$ be a convex quadrilateral with $AB = CD = 10$, $BC = 14$, and $AD = 2\sqrt{65}$. Assume that the diagonals of $ABCD$ intersect at point $P$, and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD$. Find the area of quadrilateral $ABCD$.

Diagram

Let $AP=x$ and let $PC=\rho x$. Let $[ABP]=\Delta$ and let $[ADP]=\Lambda$. [asy] size(200); defaultpen(linewidth(0.4)+fontsize(10)); pen s = linewidth(0.8)+fontsize(8); pair A, B, C, D, P; A = origin; D = (2*sqrt(65), 0); B = 10*dir(85); C = D + 10*dir(97.5); P = extension(A,C,B,D); draw(A--B--C--D--cycle, black+0.8); draw(A--C^^B--D); dot("$A$", A, SW); dot("$B$", B, NW); dot("$C$", C, NE); dot("$D$", D, SE); dot("$P$", P, S); label("$10$", A--B, W); label("$10$", C--D, E); label("$14$", B--C, N); label("$2\sqrt{65}$", A--D, S); label("$x$", A--P, SE); label("$\rho x$", P--C, SE); label("$\Delta$", B--P/2, S); label("$\Lambda$", D--P/2, 3*NW); [/asy]

Solution 1

Let $AP=x$ and let $PC=\rho x$. Let $[ABP]=\Delta$ and let $[ADP]=\Lambda$. We easily get $[PBC]=\rho \Delta$ and $[PCD]=\rho\Lambda$.

We are given that $[APB] +[CPD] = [BPC]+[APD]$, which we can now write as \[\Delta + \rho\Lambda = \rho\Delta + \Lambda \qquad \Longrightarrow \qquad \Delta -\Lambda = \rho (\Delta -\Lambda).\] Either $\Delta = \Lambda$ or $\rho=1$. The former would imply that $ABCD$ is a parallelogram, which it isn't; therefore we conclude $\rho=1$ and $P$ is the midpoint of $AC$. Let $\angle BAD = \theta$ and $\angle BCD = \phi$. Then $[ABCD]=2\cdot [BCD]=140\sin\phi$. On one hand, since $[ABD]=[BCD]$, we have \begin{align}\sqrt{65}\sin\theta = 7\sin\phi \quad \implies \quad 16+49\cos^2\phi = 65\cos^2\theta\end{align}whereas, on the other hand, using cosine formula to get the length of $BD$, we get \[10^2+4\cdot 65 - 40\sqrt{65}\cos\theta = 10^2+14^2-280\cos\phi\]\begin{align}\tag{2}\implies \qquad 65\cos^2\theta = \left(7\cos\phi+ \frac{8}{5}\right)^2\end{align}Eliminating $\cos\theta$ in the above two equations and solving for $\cos\phi$ we get\[\cos\phi = -\frac{3}{5}\qquad \implies \qquad \sin\phi = \frac{4}{5}\]which finally yields $[ABCD]=2\cdot [BCD] = 140\sin\phi = 112$.

Solution 1

For reference, $2\sqrt{65} \approx 16$, so $\overline{AD}$ is the longest of the four sides of $ABCD$. Let $h_1$ be the length of the altitude from $B$ to $\overline{AC}$, and let $h_2$ be the length of the altitude from $D$ to $\overline{AC}$. Then, the triangle area equation becomes

\[\frac{h_1}{2}AP + \frac{h_2}{2}CP = \frac{h_1}{2}CP + \frac{h_2}{2}AP \rightarrow \left(h_1 - h_2\right)AP = \left(h_1 - h_2\right)CP \rightarrow AP = CP\]

What an important finding! Note that the opposite sides $\overline{AB}$ and $\overline{CD}$ have equal length, and note that diagonal $\overline{DB}$ bisects diagonal $\overline{AC}$. This is very similar to what happens if $ABCD$ were a parallelogram with $AB = CD = 10$, so let's extend $\overline{DB}$ to point $E$, such that $AECD$ is a parallelogram. In other words, \[AE = CD = 10\] and \[EC = DA = 2\sqrt{65}\] Now, let's examine $\triangle ABE$. Since $AB = AE = 10$, the triangle is isosceles, and $\angle ABE \cong \angle AEB$. Note that in parallelogram $AECD$, $\angle AED$ and $\angle CDE$ are congruent, so $\angle ABE \cong \angle CDE$ and thus \[\text{m}\angle ABD = 180^\circ - \text{m}\angle CDB\] Define $\alpha := \text{m}\angle CDB$, so $180^\circ - \alpha = \text{m}\angle ABD$.

We use the Law of Cosines on $\triangle DAB$ and $\triangle CDB$:

\[\left(2\sqrt{65}\right)^2 = 10^2 + BD^2 - 20BD\cos\left(180^\circ - \alpha\right) = 100 + BD^2 + 20BD\cos\alpha\]

\[14^2 = 10^2 + BD^2 - 20BD\cos\alpha\]

Subtracting the second equation from the first yields

\[260 - 196 = 40BD\cos\alpha \rightarrow BD\cos\alpha = \frac{8}{5}\]

This means that dropping an altitude from $B$ to some foot $Q$ on $\overline{CD}$ gives $DQ = \frac{8}{5}$ and therefore $CQ = \frac{42}{5}$. Seeing that $CQ = \frac{3}{5}\cdot BC$, we conclude that $\triangle QCB$ is a 3-4-5 right triangle, so $BQ = \frac{56}{5}$. Then, the area of $\triangle BCD$ is $\frac{1}{2}\cdot 10 \cdot \frac{56}{5} = 56$. Since $AP = CP$, points $A$ and $C$ are equidistant from $\overline{BD}$, so \[\left[\triangle ABD\right] = \left[\triangle CBD\right] = 56\] and hence \[\left[ABCD\right] = 56 + 56 = \boxed{112}\] -kgator


Just to be complete -- $h_1$ and $h_2$ can actually be equal. In this case, $AP \neq CP$, but $BP$ must be equal to $DP$. We get the same result. -Mathdummy.

Solution 2 (Another way to get the middle point)

So, let the area of $4$ triangles $\triangle {ABP}=S_{1}$, $\triangle {BCP}=S_{2}$, $\triangle {CDP}=S_{3}$, $\triangle {DAP}=S_{4}$. Suppose $S_{1}>S_{3}$ and $S_{2}>S_{4}$, then it is easy to show that \[S_{1}\cdot S_{3}=S_{2}\cdot S_{4}.\] Also, because \[S_{1}+S_{3}=S_{2}+S_{4},\] we will have \[(S_{1}+S_{3})^2=(S_{2}+S_{4})^2.\] So \[(S_{1}+S_{3})^2=S_{1}^2+S_{3}^2+2\cdot S_{1}\cdot S_{3}=(S_{2}+S_{4})^2=S_{2}^2+S_{4}^2+2\cdot S_{2}\cdot S_{4}.\] So \[S_{1}^2+S_{3}^2=S_{2}^2+S_{4}^2.\] So \[S_{1}^2+S_{3}^2-2\cdot S_{1}\cdot S_{3}=S_{2}^2+S_{4}^2-2\cdot S_{2}\cdot S_{4}.\] So \[(S_{1}-S_{3})^2=(S_{2}-S_{4})^2.\] As a result, \[S_{1}-S_{3}=S_{2}-S_{4}.\] Then, we have \[S_{1}+S_{4}=S_{2}+S_{3}.\] Combine the condition \[S_{1}+S_{3}=S_{2}+S_{4},\] we can find out that \[S_{3}=S_{4},\] so $P$ is the midpoint of $\overline {AC}$

~Solution by $BladeRunnerAUG$ (Frank FYC)

Solution 3 (With yet another way to get the middle point)

Denote $\angle APB$ by $\alpha$. Then $\sin(\angle APB)=\sin \alpha = \sin(\angle APD)$. Using the formula for the area of a triangle, we get \[\frac{1}{2} (AP\cdot BP+ CP\cdot DP)\sin\alpha=\frac{1}{2}(AP\cdot DP+ CP\cdot BP)\sin\alpha ,\] so \[(AP-CP)(BP-DP)=0\] Hence $AP=CP$ (note that $BP=DP$ makes no difference here). Now, assume that $AP=CP=x$, $BP=y$, and $DP=z$. Using the cosine rule for $\triangle APB$ and $\triangle BPC$, it is clear that \[x^2+y^2-100=2 xy \cdot \cos{APB}=-(2 xy \cdot \cos{(\pi-CPB)})=-(x^2+y^2-196)\] or \begin{align}x^2+y^2=148\end{align}. Likewise, using the cosine rule for triangles $APD$ and $CPD$, \begin{align}\tag{2}x^2+z^2=180\end{align}. It follows that \begin{align}\tag{3}z^2-y^2=32\end{align}. Since $\sin\alpha=\sqrt{1-\cos^2\alpha}$, \[\sqrt{1-\frac{(x^2+y^2-100)^2}{4x^2y^2}}=\sqrt{1-\frac{(x^2+z^2-260)^2}{4x^2z^2}}\] which simplifies to \[\frac{48^2}{y^2}=\frac{80^2}{z^2} \qquad \Longrightarrow \qquad 5y=3z.\] Plugging this back to equations $(1)$, $(2)$, and $(3)$, it can be solved that $x=\sqrt{130},y=3\sqrt{2},z=5\sqrt{2}$. Then, the area of the quadrilateral is \[x(y+z)\sin\alpha=\sqrt{130}\cdot8\sqrt{2}\cdot\frac{14}{\sqrt{260}}=\boxed{112}\] --Solution by MicGu

Solution 4

As in all other solutions, we can first find that either $AP=CP$ or $BP=DP$, but it's an AIME problem, we can take $AP=CP$, and assume the other choice will lead to the same result (which is true).

From $AP=CP$, we have $[DAP]=[DCP]$, and $[BAP]=[BCP] \implies [ABD] = [CBD]$, therefore, \begin{align} \nonumber \frac 12 \cdot AB\cdot AD\sin A &= \frac 12 \cdot BC\cdot CD\sin C \\ \Longrightarrow \hspace{1in} 7\sin C &= \sqrt{65}\sin A \end{align} By Law of Cosines, \begin{align} \nonumber 10^2+14^2-2\cdot 10\cdot 14\cos C &= 10^2+4\cdot 65-2\cdot 10\cdot 2\sqrt{65}\cos A \\ \Longrightarrow \hspace{1in} -\frac 85 - 7\cos C &= \sqrt{65}\cos A \tag{2} \end{align} Square $(1)$ and $(2)$, and add them, to get \[\left(\frac 85\right)^2 + 2\cdot \frac 85 \cdot 7\cos C + 7^2 = 65\] Solve, $\cos C = 3/5 \implies \sin C = 4/5$, \[[ABCD] = 2[BCD] = BC\cdot CD\cdot \sin C = 14\cdot 10\cdot \frac 45 = \boxed{112}\] -Mathdummy

See Also

2018 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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