Mock AIME I 2012 Problems/Problem 3

Problem

Triangle $MNO$ has $MN=11$ , $NO=21$ , $MO=23$ . The trisection points of $MO$ are $E$ and $F$ , with $ME<MF$ . Segments $NE$ and $NF$ are extended to points $E'$ and $F'$ such that $ME' || NF$ and $OF' || NE$ . The area of pentagon $MNOF'E'$ is $p\sqrt{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ .

Solution

Use Heron's formula to find $A=[MNO]=\frac{33}{4}\sqrt{195}$ . Also note from the trisection that $[NME]=[NEF]=[NFO]=A/3$ . Now $ME'\parallel NF,ME=EF\Longrightarrow NE=EE'$ . Similarly, $NF=FF'$ . From this we deduce that

(i) $[NEM]=[MEE']\Longrightarrow [NME']=2[NME]=\frac{2}{3}A$

(ii) Similarly to (i), $[NOF']=\frac{2}{3}A$

(iii) $\triangle NEF\sim\triangle NE'F'$ with ratio $\frac{1}{2}$ , so $[NE'F']=4[NEF]=\frac{4}{3}A$

The area of our pentagon is $[NME']+[NOF']+[NE'F']=\frac{2}{3}A+\frac{2}{3}A+\frac{4}{3}A=\frac{8}{3}\cdot \frac{33}{4}\sqrt{195}=22\sqrt{195}$ . The answer is $22+195=\fbox{217}$ .

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Mock AIME I 2012 Problems/Problem 3

Problem

Solution