Difference between revisions of "2005 AIME I Problems/Problem 15"

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Triangle <math> ABC </math> has <math> BC=20. </math> The [[incircle]] of the triangle evenly [[trisect]]s the [[median of a triangle | median]] <math> AD. </math> If the area of the triangle is <math> m \sqrt{n} </math> where <math> m </math> and <math> n </math> are integers and <math> n </math> is not [[divisor | divisible]] by the [[perfect square | square]] of a prime, find <math> m+n. </math>
 
Triangle <math> ABC </math> has <math> BC=20. </math> The [[incircle]] of the triangle evenly [[trisect]]s the [[median of a triangle | median]] <math> AD. </math> If the area of the triangle is <math> m \sqrt{n} </math> where <math> m </math> and <math> n </math> are integers and <math> n </math> is not [[divisor | divisible]] by the [[perfect square | square]] of a prime, find <math> m+n. </math>
  
== Solution ==
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== Solution 1==
 
<center><asy>
 
<center><asy>
 
size(300); pointpen=black;pathpen=black+linewidth(0.65); pen s = fontsize(10);
 
size(300); pointpen=black;pathpen=black+linewidth(0.65); pen s = fontsize(10);
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Thus <math>c = 2</math> or <math> = 10</math>.  We discard the value <math>c = 10</math> as extraneous (it gives us an [[equilateral triangle]]) and are left with <math>c = 2</math>, so our triangle has area <math>\sqrt{28 \cdot 18 \cdot 8 \cdot 2} = 24\sqrt{14}</math> and so the answer is <math>24 + 14 = \boxed{038}</math>.
 
Thus <math>c = 2</math> or <math> = 10</math>.  We discard the value <math>c = 10</math> as extraneous (it gives us an [[equilateral triangle]]) and are left with <math>c = 2</math>, so our triangle has area <math>\sqrt{28 \cdot 18 \cdot 8 \cdot 2} = 24\sqrt{14}</math> and so the answer is <math>24 + 14 = \boxed{038}</math>.
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== Solution 2 ==
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Use Power of a point similar to the first solution to find that <math>AB = 10</math> and that the side <math>AC = 2 \cdot x \sqrt{2}</math>, where <math>x</math> is one third of the median's length. Then use systems of law of cosines, creating two triangles, with <math>10-10-3x</math> with angle <math>\theta</math>, and <math>10-20-2 \cdot x \sqrt{2}</math> with the same angle. Solving the system yields <math>x = 4 \sqrt{2}</math>. Solving using Heron's Formula gets the answer <math>24 \sqrt{14}</math>, or <math>\boxed{038}</math>.
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== See also ==
 
== See also ==

Revision as of 00:02, 27 July 2018

Problem

Triangle $ABC$ has $BC=20.$ The incircle of the triangle evenly trisects the median $AD.$ If the area of the triangle is $m \sqrt{n}$ where $m$ and $n$ are integers and $n$ is not divisible by the square of a prime, find $m+n.$

Solution 1

[asy] size(300); pointpen=black;pathpen=black+linewidth(0.65); pen s = fontsize(10); pair A=(0,0),B=(26,0),C=IP(circle(A,10),circle(B,20)),D=(B+C)/2,I=incenter(A,B,C); path cir = incircle(A,B,C); pair E1=IP(cir,B--C),F=IP(cir,A--C),G=IP(cir,A--B),P=IP(A--D,cir),Q=OP(A--D,cir); D(MP("A",A,s)--MP("B",B,s)--MP("C",C,N,s)--cycle); D(cir);  D(A--MP("D",D,NE,s)); D(MP("E",E1,NE,s)); D(MP("F",F,NW,s)); D(MP("G",G,s)); D(MP("P",P,SW,s)); D(MP("Q",Q,SE,s)); MP("10",(B+D)/2,NE); MP("10",(C+D)/2,NE); [/asy]

Let $E$, $F$ and $G$ be the points of tangency of the incircle with $BC$, $AC$ and $AB$, respectively. Without loss of generality, let $AC < AB$, so that $E$ is between $D$ and $C$. Let the length of the median be $3m$. Then by two applications of the Power of a Point Theorem, $DE^2 = 2m \cdot m = AF^2$, so $DE = AF$. Now, $CE$ and $CF$ are two tangents to a circle from the same point, so $CE = CF = c$ and thus $AC = AF + CF = DE + CE = CD = 10$. Then $DE = AF = AG = 10 - c$ so $BG = BE = BD + DE = 20 - c$ and thus $AB = AG + BG = 30 - 2c$.

Now, by Stewart's Theorem in triangle $\triangle ABC$ with cevian $\overline{AD}$, we have

\[(3m)^2\cdot 20 + 20\cdot10\cdot10 = 10^2\cdot10 + (30 - 2c)^2\cdot 10.\]

Our earlier result from Power of a Point was that $2m^2 = (10 - c)^2$, so we combine these two results to solve for $c$ and we get

\[9(10 - c)^2 + 200 = 100 + (30 - 2c)^2 \quad \Longrightarrow \quad c^2 - 12c + 20 = 0.\]

Thus $c = 2$ or $= 10$. We discard the value $c = 10$ as extraneous (it gives us an equilateral triangle) and are left with $c = 2$, so our triangle has area $\sqrt{28 \cdot 18 \cdot 8 \cdot 2} = 24\sqrt{14}$ and so the answer is $24 + 14 = \boxed{038}$.

Solution 2

Use Power of a point similar to the first solution to find that $AB = 10$ and that the side $AC = 2 \cdot x \sqrt{2}$, where $x$ is one third of the median's length. Then use systems of law of cosines, creating two triangles, with $10-10-3x$ with angle $\theta$, and $10-20-2 \cdot x \sqrt{2}$ with the same angle. Solving the system yields $x = 4 \sqrt{2}$. Solving using Heron's Formula gets the answer $24 \sqrt{14}$, or $\boxed{038}$.


See also

2005 AIME I (ProblemsAnswer KeyResources)
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Problem 14
Followed by
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All AIME Problems and Solutions

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