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Difference between revisions of "2002 AIME I Problems/Problem 13"

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== Problem ==
 
== Problem ==
−
In triangle <math>ABC</math> the medians <math>\overline{AD}</math> and <math>\overline{CE}</math> have lengths 18 and 27, respectively, and <math>AB=24</math>. Extend <math>\overline{CE}</math> to intersect the circumcircle of <math>ABC</math> at <math>F</math>. The area of triangle <math>AFB</math> is <math>m\sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers and <math>n</math> is not divisible by the square of any prime. Find m+n.
+
In [[triangle]] <math>ABC</math> the [[median]]s <math>\overline{AD}</math> and <math>\overline{CE}</math> have lengths <math>18</math> and <math>27</math>, respectively, and <math>AB=24</math>. Extend <math>\overline{CE}</math> to intersect the [[circumcircle]] of <math>ABC</math> at <math>F</math>. The area of triangle <math>AFB</math> is <math>m\sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers and <math>n</math> is not divisible by the square of any prime. Find <math>m+n</math>.
  
−
== Solution ==
+
== Solution 1==
−
{{solution}}
+
<center><asy>
 +
size(150); pathpen = linewidth(0.7); pointpen = black; pen f = fontsize(8); pair A=(0,0), B=(24,0), E=(A+B)/2, C=IP(CR(A,3*70^.5),CR(E,27)), D=(B+C)/2, F=IP(circumcircle(A,B,C),E--C+2*(E-C));
 +
D(D(MP("A",A))--D(MP("B",B))--D(MP("C",C,NW))--cycle); D(circumcircle(A,B,C)); D(MP("F",F)); D(A--D); D(C--F); D(A--F--B); D(MP("E",E,NE)); D(MP("D",D,NE)); MP("12",(A+E)/2,SE,f);MP("12",(B+E)/2,f); MP("27",(C+E)/2,SW,f); MP("18",(A+D)/2,SE,f);
 +
</asy></center>
 +
 
 +
Applying [[Stewart's Theorem]] to medians <math>AD, CE</math>, we have:
 +
<center>
 +
<cmath>\begin{align*}
 +
BC^2 + 4 \cdot 18^2 &= 2\left(24^2 + AC^2\right) \\
 +
24^2 + 4 \cdot 27^2 &= 2\left(AC^2 + BC^2\right)
 +
\end{align*}</cmath>
 +
</center>
 +
Substituting the first equation into the second and simplification yields <math>24^2 = 2\left(3AC^2 + 2 \cdot 24^2 - 4 \cdot 18^2\right)- 4 \cdot 27^2</math> <math> \Longrightarrow AC = \sqrt{2^5 \cdot 3 + 2 \cdot 3^5 + 2^4 \cdot 3^3 - 2^7 \cdot 3} = 3\sqrt{70}</math>.
 +
 
 +
By the [[Power of a Point Theorem]] on <math>E</math>, we get <math>EF = \frac{12^2}{27} = \frac{16}{3}</math>. The [[Law of Cosines]] on <math>\triangle ACE</math> gives
 +
<center>
 +
<cmath>\begin{align*}
 +
\cos \angle AEC = \left(\frac{12^2 + 27^2 - 9 \cdot 70}{2 \cdot 12 \cdot 27}\right) = \frac{3}{8}
 +
\end{align*}</cmath>
 +
</center>
 +
Hence <math>\sin \angle AEC = \sqrt{1 - \cos^2 \angle AEC} = \frac{\sqrt{55}}{8}</math>. Because <math>\triangle AEF, BEF</math> have the same height and equal bases, they have the same area, and <math>[ABF] = 2[AEF] = 2 \cdot \frac 12 \cdot AE \cdot EF \sin \angle AEF = 12 \cdot \frac{16}{3} \cdot \frac{\sqrt{55}}{8} = 8\sqrt{55}</math>, and the answer is <math>8 + 55 = \boxed{063}</math>.
 +
 
 +
== Solution 2 ==
 +
 
 +
Let <math>AD</math> and <math>CE</math> intersect at <math>P</math>. Since medians split one another in a 2:1 ratio, we have
 +
<center>
 +
<cmath>\begin{align*}
 +
AP = 12, PE = 9
 +
\end{align*}</cmath>
 +
</center>
 +
This gives isosceles <math>APE</math> and thus an easy area calculation. After extending the altitude to <math>PE</math> and using the fact that it is also a median, we find
 +
<center>
 +
<cmath>\begin{align*}
 +
[APE] = \frac{27\sqrt{55}}{4}
 +
\end{align*}</cmath>
 +
</center>
 +
Using Power of a Point, we have
 +
<center>
 +
<cmath>\begin{align*}
 +
EF=\frac{16}{3}
 +
\end{align*}</cmath>
 +
</center>
 +
By Same Height Different Base,
 +
<center>
 +
<cmath>\begin{align*}
 +
\frac{[AFE]}{[APE]}=\frac{[AFE]}{(\frac{27\sqrt{55}}{4})}=\frac{EF}{PE}=\frac{(\frac{16}{3})}{9}=\frac{16}{27}
 +
\end{align*}</cmath>
 +
</center>
 +
Solving gives
 +
<center>
 +
<cmath>\begin{align*}
 +
[AFE] = 4\sqrt{55}
 +
\end{align*}</cmath>
 +
</center>
 +
and
 +
<center>
 +
<cmath>\begin{align*}
 +
[AFB]=2[AFE]=8\sqrt{55}
 +
\end{align*}</cmath>
 +
</center>
 +
Thus, our answer is <math>8+55=\boxed{063}</math>.
 +
 
 +
==Short Solution: Smart Similarity==
 +
 
 +
Use the same diagram as in Solution 1. Call the centroid <math>P</math>. It should be clear that <math>PE=9</math>, and likewise <math>AP=12</math>, <math>AE=12</math>. Then, <math>\sin \angle AEP = \frac{\sqrt{55}}{8}</math>. Power of a Point on <math>E</math> gives <math>FE=\frac{16}{3}</math>, and the area of <math>AFB</math> is <math>AE * EF* \sin \angle AEP</math>, which is twice the area of <math>AEF</math> or <math>FEB</math> (they have the same area because of equal base and height), giving <math>8\sqrt{55}</math> for an answer of <math>\boxed{063}</math>.
 +
 
 +
== Solution 4 (You've Forgotten Power of a Point Exists) ==
 +
 
 +
Note that, as above, it is quite easy to get that <math>\sin \angle AEP = \frac{\sqrt{55}}{8}</math> (equate Heron's and <math>\frac{1}{2}ab\sin C</math> to find this). Now note that <math>\angle FEA = \angle BEC</math> because they are vertical angles, <math>\angle FAE = \angle ECB</math>, and <math>\angle EFA = \angle ABC</math> (the latter two are derived from the inscribed angle theorem). Therefore <math>\Delta AEF </math> ~ <math> \Delta CEB</math> and so <math>FE = \frac{144}{27}</math> and <math>\sin \angle FEA = \frac{\sqrt{55}}{8}</math> so the area of <math>\Delta BFA</math> is <math>8\sqrt{55}</math> giving us <math>\boxed{063}</math> as our answer. (One may just get the area via triangle similarity too--this is if you are tired by the end of test and just want to bash some stuff out--it may also serve as a useful check).
 +
 
 +
~Dhillonr25
 +
 
 +
== Solution 5 (Barycentric Coordinates) ==
 +
Apply barycentric coordinates on <math>\triangle ABC</math>. We know that <math>D=\left(0, \frac{1}{2}, \frac{1}{2}\right), E=\left(\frac{1}{2}, \frac{1}{2}, 0\right)</math>. We can now get the displacement vectors <math>\overrightarrow{AD} = \left(1, -\frac{1}{2}, -\frac{1}{2}\right)</math> and <math>\overrightarrow{CE}=\left(-\frac{1}{2}, -\frac{1}{2}, 1\right)</math>. Now, applying the distance formula and simplifying gives us the two equations
 +
<cmath>\begin{align*}
 +
2b^2+2c^2-a^2&=1296 \\
 +
2a^2+2b^2-c^2&=2916. \\
 +
\end{align*}</cmath>
 +
Substituting <math>c=24</math> and solving with algebra now gives <math>a=6\sqrt{31}, b=3\sqrt{70}</math>. Now we can find <math>F</math>. Note that <math>CE</math> can be parameterized as <math>(1:1:t)</math>, so plugging into the circumcircle equation and solving for <math>t</math> gives <math>t=\frac{-c^2}{a^2+b^2}</math> so <math>F=(a^2+b^2:a^2+b^2:-c^2)</math>. Plugging in for <math>a,b</math> gives us <math>F=(1746:1746:-576)</math>. Thus, by the area formula, we have<cmath>\frac{[AFB]}{[ABC]}=
 +
\left|\begin{matrix}
 +
1 & 0 & 0 \\
 +
0 & 1 & 0 \\
 +
\frac{97}{162} & \frac{97}{162} & -\frac{16}{81}
 +
\end{matrix}\right|=\frac{16}{81}.</cmath>By Heron's Formula, we have <math>[ABC]=\frac{81\sqrt{55}}{2}</math> which immediately gives <math>[AFB]=8\sqrt{55}</math> from our ratio, extracting <math>\boxed{063}</math>.
 +
 
 +
-Taco12
 +
 
 +
==Solution 6 (Law of Cosines + Stewarts)==
 +
Since <math>AD</math> is the median, let <math>BD=BC=x</math>. Since <math>CE</math> is a median, <math>AE=BE=12</math>. Applying Power of a Point with respect to point <math>E</math>, we see that <math>EF=\frac{16}{3}</math>. Applying Stewart's Theorem on triangles <math>\triangle ADC</math> and <math>\triangle ABC</math>, we get that <math>x=3\sqrt{31}</math> and <math>y=3\sqrt{70}</math>. The area of <math>\triangle AFB</math> is simply <math>\frac{1}{2} \cdot \sin{\angle FBA}\cdot FB\cdot AB</math>. We know <math>AB=24</math>. Also, we know that <math>\angle FBA = \angle FCA</math>. Then, applying Law of Cosines on triangle <math>EAC</math>, we get that <math>\cos{\angle FCA}=\frac{3\sqrt{70}}{28}</math> which means that <math>\sin{\angle FCA=\angle FBA}=\frac{\sqrt{154}}{28}</math>. Then, applying Stewart's Theorem on triangle <math>FBC</math> with cevian <math>BE</math> allows us to receive that <math>FB=\frac{4\sqrt{70}}{3}</math>. Now, plugging into our earlier area formula, we receive <math>\frac{1}{2} \cdot \frac{\sqrt{154}}{28} \cdot \frac{4\sqrt{70}}{3} \cdot 24 = 8\sqrt{55}.</math> Therefore, the desired answer is <math>8+55=\boxed{063}</math>.
 +
 
 +
~SirAppel
  
 
== See also ==
 
== See also ==
−
* [[2002 AIME I Problems/Problem 12| Previous problem]]
+
{{AIME box|year=2002|n=I|num-b=12|num-a=14}}
−
 
 
−
* [[2002 AIME I Problems/Problem 14| Next problem]]
 
  
−
* [[2002 AIME I Problems]]
+
[[Category:Intermediate Geometry Problems]]
 +
{{MAA Notice}}

Latest revision as of 22:15, 11 July 2024

Problem

In triangle $ABC$ the medians $\overline{AD}$ and $\overline{CE}$ have lengths $18$ and $27$, respectively, and $AB=24$. Extend $\overline{CE}$ to intersect the circumcircle of $ABC$ at $F$. The area of triangle $AFB$ is $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n$.

Solution 1

[asy] size(150); pathpen = linewidth(0.7); pointpen = black; pen f = fontsize(8); pair A=(0,0), B=(24,0), E=(A+B)/2, C=IP(CR(A,3*70^.5),CR(E,27)), D=(B+C)/2, F=IP(circumcircle(A,B,C),E--C+2*(E-C)); D(D(MP("A",A))--D(MP("B",B))--D(MP("C",C,NW))--cycle); D(circumcircle(A,B,C)); D(MP("F",F)); D(A--D); D(C--F); D(A--F--B); D(MP("E",E,NE)); D(MP("D",D,NE)); MP("12",(A+E)/2,SE,f);MP("12",(B+E)/2,f); MP("27",(C+E)/2,SW,f); MP("18",(A+D)/2,SE,f); [/asy]

Applying Stewart's Theorem to medians $AD, CE$, we have:

\begin{align*} BC^2 + 4 \cdot 18^2 &= 2\left(24^2 + AC^2\right) \\ 24^2 + 4 \cdot 27^2 &= 2\left(AC^2 + BC^2\right) \end{align*}

Substituting the first equation into the second and simplification yields $24^2 = 2\left(3AC^2 + 2 \cdot 24^2 - 4 \cdot 18^2\right)- 4 \cdot 27^2$ $\Longrightarrow AC = \sqrt{2^5 \cdot 3 + 2 \cdot 3^5 + 2^4 \cdot 3^3 - 2^7 \cdot 3} = 3\sqrt{70}$.

By the Power of a Point Theorem on $E$, we get $EF = \frac{12^2}{27} = \frac{16}{3}$. The Law of Cosines on $\triangle ACE$ gives

\begin{align*} \cos \angle AEC = \left(\frac{12^2 + 27^2 - 9 \cdot 70}{2 \cdot 12 \cdot 27}\right) = \frac{3}{8} \end{align*}

Hence $\sin \angle AEC = \sqrt{1 - \cos^2 \angle AEC} = \frac{\sqrt{55}}{8}$. Because $\triangle AEF, BEF$ have the same height and equal bases, they have the same area, and $[ABF] = 2[AEF] = 2 \cdot \frac 12 \cdot AE \cdot EF \sin \angle AEF = 12 \cdot \frac{16}{3} \cdot \frac{\sqrt{55}}{8} = 8\sqrt{55}$, and the answer is $8 + 55 = \boxed{063}$.

Solution 2

Let $AD$ and $CE$ intersect at $P$. Since medians split one another in a 2:1 ratio, we have

\begin{align*} AP = 12, PE = 9 \end{align*}

This gives isosceles $APE$ and thus an easy area calculation. After extending the altitude to $PE$ and using the fact that it is also a median, we find

\begin{align*} [APE] = \frac{27\sqrt{55}}{4} \end{align*}

Using Power of a Point, we have

\begin{align*} EF=\frac{16}{3} \end{align*}

By Same Height Different Base,

\begin{align*} \frac{[AFE]}{[APE]}=\frac{[AFE]}{(\frac{27\sqrt{55}}{4})}=\frac{EF}{PE}=\frac{(\frac{16}{3})}{9}=\frac{16}{27} \end{align*}

Solving gives

\begin{align*} [AFE] = 4\sqrt{55} \end{align*}

and

\begin{align*} [AFB]=2[AFE]=8\sqrt{55} \end{align*}

Thus, our answer is $8+55=\boxed{063}$.

Short Solution: Smart Similarity

Use the same diagram as in Solution 1. Call the centroid $P$. It should be clear that $PE=9$, and likewise $AP=12$, $AE=12$. Then, $\sin \angle AEP = \frac{\sqrt{55}}{8}$. Power of a Point on $E$ gives $FE=\frac{16}{3}$, and the area of $AFB$ is $AE * EF* \sin \angle AEP$, which is twice the area of $AEF$ or $FEB$ (they have the same area because of equal base and height), giving $8\sqrt{55}$ for an answer of $\boxed{063}$.

Solution 4 (You've Forgotten Power of a Point Exists)

Note that, as above, it is quite easy to get that $\sin \angle AEP = \frac{\sqrt{55}}{8}$ (equate Heron's and $\frac{1}{2}ab\sin C$ to find this). Now note that $\angle FEA = \angle BEC$ because they are vertical angles, $\angle FAE = \angle ECB$, and $\angle EFA = \angle ABC$ (the latter two are derived from the inscribed angle theorem). Therefore $\Delta AEF$ ~ $\Delta CEB$ and so $FE = \frac{144}{27}$ and $\sin \angle FEA = \frac{\sqrt{55}}{8}$ so the area of $\Delta BFA$ is $8\sqrt{55}$ giving us $\boxed{063}$ as our answer. (One may just get the area via triangle similarity too--this is if you are tired by the end of test and just want to bash some stuff out--it may also serve as a useful check).

~Dhillonr25

Solution 5 (Barycentric Coordinates)

Apply barycentric coordinates on $\triangle ABC$. We know that $D=\left(0, \frac{1}{2}, \frac{1}{2}\right), E=\left(\frac{1}{2}, \frac{1}{2}, 0\right)$. We can now get the displacement vectors $\overrightarrow{AD} = \left(1, -\frac{1}{2}, -\frac{1}{2}\right)$ and $\overrightarrow{CE}=\left(-\frac{1}{2}, -\frac{1}{2}, 1\right)$. Now, applying the distance formula and simplifying gives us the two equations \begin{align*} 2b^2+2c^2-a^2&=1296 \\ 2a^2+2b^2-c^2&=2916. \\ \end{align*} Substituting $c=24$ and solving with algebra now gives $a=6\sqrt{31}, b=3\sqrt{70}$. Now we can find $F$. Note that $CE$ can be parameterized as $(1:1:t)$, so plugging into the circumcircle equation and solving for $t$ gives $t=\frac{-c^2}{a^2+b^2}$ so $F=(a^2+b^2:a^2+b^2:-c^2)$. Plugging in for $a,b$ gives us $F=(1746:1746:-576)$. Thus, by the area formula, we have\[\frac{[AFB]}{[ABC]}= \left|\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ \frac{97}{162} & \frac{97}{162} & -\frac{16}{81} \end{matrix}\right|=\frac{16}{81}.\]By Heron's Formula, we have $[ABC]=\frac{81\sqrt{55}}{2}$ which immediately gives $[AFB]=8\sqrt{55}$ from our ratio, extracting $\boxed{063}$.

-Taco12

Solution 6 (Law of Cosines + Stewarts)

Since $AD$ is the median, let $BD=BC=x$. Since $CE$ is a median, $AE=BE=12$. Applying Power of a Point with respect to point $E$, we see that $EF=\frac{16}{3}$. Applying Stewart's Theorem on triangles $\triangle ADC$ and $\triangle ABC$, we get that $x=3\sqrt{31}$ and $y=3\sqrt{70}$. The area of $\triangle AFB$ is simply $\frac{1}{2} \cdot \sin{\angle FBA}\cdot FB\cdot AB$. We know $AB=24$. Also, we know that $\angle FBA = \angle FCA$. Then, applying Law of Cosines on triangle $EAC$, we get that $\cos{\angle FCA}=\frac{3\sqrt{70}}{28}$ which means that $\sin{\angle FCA=\angle FBA}=\frac{\sqrt{154}}{28}$. Then, applying Stewart's Theorem on triangle $FBC$ with cevian $BE$ allows us to receive that $FB=\frac{4\sqrt{70}}{3}$. Now, plugging into our earlier area formula, we receive $\frac{1}{2} \cdot \frac{\sqrt{154}}{28} \cdot \frac{4\sqrt{70}}{3} \cdot 24 = 8\sqrt{55}.$ Therefore, the desired answer is $8+55=\boxed{063}$.

~SirAppel

See also

2002 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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