Difference between revisions of "2011 AIME I Problems/Problem 4"

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Problem 4

In triangle $ABC$ , $AB=125$ , $AC=117$ and $BC=120$ . The angle bisector of angle $A$ intersects $\overline{BC}$ at point $L$ , and the angle bisector of angle $B$ intersects $\overline{AC}$ at point $K$ . Let $M$ and $N$ be the feet of the perpendiculars from $C$ to $\overline{BK}$ and $\overline{AL}$ , respectively. Find $MN$ .

Solution

Solution 1

Extend ${MN}$ such that it intersects lines ${AC}$ and ${BC}$ at points $O$ and $Q$ , respectively.

Lemma 1: $O, Q$ are midpoints of $AC$ and $BC$

Proof: Consider the reflection of the vertex $C$ over the line $BM$ , and let this point be $C_1$ . Since $\angle{BMC} = 90^{\circ}$ , we have that $C_1$ is the image of $C$ after reflection over $M$ , and from the definition of reflection $\angle{MBC} = \angle{MBC_1}$ . Then it is easily seen that since $BM$ is an angle bisector, that $\angle{MBC_1} = \angle{MBA}$ , so $C_1$ lies on $AB$ . Similarly, if we define $C_2$ to be the reflection of $C$ over $N$ , then we find that $C_2$ lies on $AB$ . Then we can now see that $\triangle{CMN} \sim \triangle{CC_1C_2}$ , with a homothety of ratio $2$ taking the first triangle to the second. Then this same homothety takes everything on the line $MN$ to everything on the line $AB$ . So since $O, Q$ lie on $MN$ , this homothety also takes $O, Q$ to $A, B$ so they are midpoints, as desired. $\Box$

Lemma 2: $\triangle{MQC}, \triangle{NOC}$ are isosceles triangles

Proof: To show that $\triangle{MQC}$ is isosceles, note that $\triangle{MQC} \sim \triangle{C_1BC}$ , with similarity ratio of $\frac{1}{2}$ . So it suffices to show that triangle $\triangle{C_1BC}$ is isosceles. But this follows quickly from Lemma 1, since $BM$ is both an altitude and an angle bisector of $\angle{C_1BC}$ . $\triangle{NOC}$ is isosceles by the same reasoning. $\Box$

Since ${OQ}$ is a midline, it then follows that ${OC} = 58.5$ and ${QC} = 60$ . Since $\triangle MQC$ and $\triangle NOC$ are both isosceles, we have that $ON = OC = 58.5$ and $MQ = QC = 60$ . Since $OQ$ is a midline, $OQ = 62.5$ . We want to find $MN$ , which is just $ON + MQ - OQ$ .

Substituting the values of $ON, MQ, OQ$ , we have that the answer is $58.5 + 60 - 62.5 = \boxed {56}$ .

Solution 2

Let $I$ be the intersection of $AL$ and $BK$ , or rather the incenter of triangle $ABC$ . Noting that $\angle IMC$ and $\angle CNI$ are right, we conclude that $CNIM$ is a cyclic quadrilateral, so by Ptolemy's Theorem, $CI\cdot MN=IM\cdot CN+NI\cdot MC.$ Now let $IP$ and $IQ$ be inradii to $AC$ and $BC$ respectively in the following picture, which is not to scale.

$[asy] size(200); pair A=(0,0),C=(117,0), B=(100,85),I=incenter(A,B,C),K=extension(B,I,A,C),L=extension(A,I,B,C),M=foot(C,B,K),EN=foot(C,A,L); D(MP("A",A)--MP("B",B,N)--MP("C",C)--cycle,black); draw(B--M--C--EN--A); draw(M--K^^EN--L); MP("M",M,WNW);MP("N",EN,NNW);MP("L",L,ENE);MP("K",K,S);MP("I",I,NW); markscalefactor=.75; draw(rightanglemark(C,M,I)^^rightanglemark(I,EN,C)); pair FAC=foot(I,A,C); pair FBC=foot(I,B,C); MP("P",FAC,S);MP("Q",FBC,ESE); draw(I--FAC^^I--FBC,dotted); draw(C--I); draw(rightanglemark(I,FAC,A)); dot(A);dot(B);dot(C);dot(M);dot(EN);dot(K);dot(L);dot(FAC);dot(FBC);dot(I); [/asy]$

We know that $\frac{\angle A+\angle B+\angle C}{2}=90^\circ$ . In triangle $CBM$ , we have $90^\circ=\angle CBM+\angle BCI+\angle ICM=\frac{\angle B}{2}+\frac{\angle C}{2}+\angle ICM.$ Therefore, $\angle ICM=\frac{\angle A}{2}$ , and $\triangle CIM\sim\triangle AIP$ . Thus $MC=CI\cdot \frac{AP}{AI}$ . Using a similar method, we can find that $CN=CI\cdot \frac{BQ}{BI}$ . Therefore, our Ptolemy's expression simplifies to $MN=IM\cdot \frac{BQ}{BI}+NI\cdot\frac{AP}{AI}=IM\cdot\cos \frac{B}{2}+NI\cdot\cos\frac{A}{2}.$ Using right triangles $CIM$ and $NCI$ , we also know that $IM=CI\cdot\sin \frac{A}{2}$ and $NI=CI\cdot\sin\frac{B}{2}$ . Thus $MN=CI\cdot\cos \frac{B}{2}\sin\frac{A}{2}+CI\cdot\cos\frac{A}{2}\sin\frac{B}{2}=CI\cdot\sin\left(\frac{A+B}{2}\right)=CI\cdot \cos \frac{C}{2}.$ But this last expression is equal to $CQ$ . This a tangent to the incircle, so it has length $s-c=181-125=\boxed{56}$ .

@@ Line 47: / Line 47: @@
 </center>
 We know that <math>\frac{\angle A+\angle B+\angle C}{2}=90^\circ</math>. In triangle <math>CBM</math>, we have <cmath>90^\circ=\angle CBM+\angle BCI+\angle ICM=\frac{\angle B}{2}+\frac{\angle C}{2}+\angle ICM.</cmath> Therefore, <math>\angle ICM=\frac{\angle A}{2}</math>, and <math>\triangle CIM\sim\triangle AIP</math>.
-Thus <math>IM=CI\cdot \frac{IP}{AI}</math>. Using a similar method, we can find that <math>NI=CI\cdot\frac{IQ}{BI}</math>. Therefore, our Ptolemy's expression simplifies to <cmath>MN=\frac{IP}{AI}\cdot CN+\frac{IQ}{BI}\cdot MC=r\left(\frac{CN}{AI}+\frac{MC}{BI}\right),</cmath>
+Thus <math>MC=CI\cdot \frac{AP}{AI}</math>. Using a similar method, we can find that <math>CN=CI\cdot \frac{BQ}{BI}</math>. Therefore, our Ptolemy's expression simplifies to <cmath>MN=IM\cdot \frac{BQ}{BI}+NI\cdot\frac{AP}{AI}=IM\cdot\cos  \frac{B}{2}+NI\cdot\cos\frac{A}{2}.</cmath>
-where <math>r</math> is the inradius of triangle <math>ABC</math>. Thus, <math>r=[ABC]/181</math>. Also, right triangles <math>CNA</math> and <math>CBM</math> tell us that <math>CN=117\sin \frac{A}{2}</math> and <math>MC=120\sin\frac{B}{2}</math>. But then <math>[CLA]=\frac{117\cdot AL}{2}\sin\frac{A}{2}</math>, and this is equal to <math>\frac{117}{242}\cdot [ABC]</math> by the Angle Bisector Theorem.  Therefore, solving this for <math>\sin\frac{A}{2}</math> and substituting yields <math>CN=\frac{2\cdot 117\cdot [ABC]}{242\cdot AL}</math>. Similarly, <math>MC=\frac{2\cdot 120\cdot [ABC]}{245\cdot BK}</math>. We now replace these in our Ptolemy's expression to get
+Using right triangles <math>CIM</math> and <math>NCI</math>, we also know that <math>IM=CI\cdot\sin \frac{A}{2}</math> and <math>NI=CI\cdot\sin\frac{B}{2}</math>. Thus <cmath>MN=CI\cdot\cos  \frac{B}{2}\sin\frac{A}{2}+CI\cdot\cos\frac{A}{2}\sin\frac{B}{2}=CI\cdot\sin\left(\frac{A+B}{2}\right)=CI\cdot \cos \frac{C}{2}.</cmath>
-<cmath>MN=\frac{[ABC]}{181}\left(\frac{2\cdot 117\cdot [ABC]}{242\cdot AL\cdot AI}+\frac{2\cdot 120\cdot [ABC]}{245\cdot BK\cdot BI}\right)</cmath><cmath>=\frac{2[ABC]^2}{181}\left(\frac{117}{242\cdot AL\cdot AI}+\frac{120}{245\cdot BK\cdot BI}\right).</cmath>
+But this last expression is equal to <math>CQ</math>. This a tangent to the incircle, so it has length <math>s-c=181-125=\boxed{56}</math>.
-We can also use [[mass points]], assigning masses of <math>120</math>, <math>117</math>, and <math>125</math> to points <math>A</math>, <math>B</math>, and <math>C</math>, respectively. Then point <math>L</math> has a mass of <math>242</math> and point <math>K</math> has a mass of <math>245</math>, so <math>AI=\frac{242}{362}\cdot AL</math> and <math>BI=\frac{245}{362}\cdot BK</math>. This simplifies our expression further to
-<cmath>MN=4[ABC]^2\left(\frac{117}{242^2\cdot AL^2}+\frac{120}{245^2\cdot BK^2}\right).</cmath>
-Then using the angle bisector formula, we find that <math>AL^2=125\cdot 117\left(1-\frac{120^2}{242^2}\right)</math> and <math>BK^2=125\cdot 120\left(1-\frac{117^2}{245^2}\right)</math>. Also, Heron's Formula tells us that <cmath>[ABC]^2=181\cdot 61\cdot 56\cdot 64,</cmath>
-so when we substitute this all in, we get <cmath>MN=4\cdot 181\cdot 61\cdot 56\cdot 64\cdot\left(\frac{1}{125(242^2-120^2)}+\frac{1}{125(245^2-117^2)}\right)</cmath><cmath>=\frac{4\cdot 181\cdot 61\cdot 56\cdot 64}{125}\cdot\left(\frac{1}{122\cdot362}+\frac{1}{128\cdot362}\right)</cmath><cmath>=\frac{2\cdot 61\cdot 56\cdot 64}{125}\cdot\left(\frac{250}{122\cdot 128}\right)=\boxed{56}.</cmath>
 == See also ==
 {{AIME box|year=2011|n=I|num-b=3|num-a=5}}
 {{MAA Notice}}

2011 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
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