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Difference between revisions of "2025 AIME I Problems/Problem 2"
(→‎Problem)
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==Problem==
 
==Problem==
 
In <math>\triangle ABC</math> points <math>D</math> and <math>E</math> lie on <math>\overline{AB}</math> so that <math>AD < AE < AB</math>, while points <math>F</math> and <math>G</math> lie on <math>\overline{AC}</math> so that <math>AF < AG < AC</math>. Suppose <math>AD = 4</math>, <math>DE = 16</math>, <math>EB = 8</math>, <math>AF = 13</math>, <math>FG = 52</math>, and <math>GC = 26</math>. Let <math>M</math> be the reflection of <math>D</math> through <math>F</math>, and let <math>N</math> be the reflection of <math>G</math> through <math>E</math>. The area of quadrilateral <math>DEGF</math> is <math>288</math>. Find the area of heptagon <math>AFNBCEM</math>, as shown in the figure below.
 
In <math>\triangle ABC</math> points <math>D</math> and <math>E</math> lie on <math>\overline{AB}</math> so that <math>AD < AE < AB</math>, while points <math>F</math> and <math>G</math> lie on <math>\overline{AC}</math> so that <math>AF < AG < AC</math>. Suppose <math>AD = 4</math>, <math>DE = 16</math>, <math>EB = 8</math>, <math>AF = 13</math>, <math>FG = 52</math>, and <math>GC = 26</math>. Let <math>M</math> be the reflection of <math>D</math> through <math>F</math>, and let <math>N</math> be the reflection of <math>G</math> through <math>E</math>. The area of quadrilateral <math>DEGF</math> is <math>288</math>. Find the area of heptagon <math>AFNBCEM</math>, as shown in the figure below.
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[asy]
 
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unitsize(14);
 
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pair A = (0, 9), B = (-6, 0), C = (12, 0), D = (5A + 2B)/7, E = (2A + 5B)/7, F = (5A + 2C)/7, G = (2A + 5C)/7, M = 2F - D, N = 2E - G;
 
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filldraw(A--F--N--B--C--E--M--cycle, lightgray);
 
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draw(A--B--C--cycle);
 
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draw(D--M);
 
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draw(N--G);
 
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dot(A);
 
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dot(B);
 
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dot(C);
 
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dot(D);
 
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dot(E);
 
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dot(F);
 
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dot(G);
 
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dot(M);
 
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dot(N);
 
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label("<math>A</math>", A, dir(90));
 
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label("<math>B</math>", B, dir(225));
 
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label("<math>C</math>", C, dir(315));
 
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label("<math>D</math>", D, dir(135));
 
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label("<math>E</math>", E, dir(135));
 
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label("<math>F</math>", F, dir(45));
 
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label("<math>G</math>", G, dir(45));
 
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label("<math>M</math>", M, dir(45));
 
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label("<math>N</math>", N, dir(135));
 
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[/asy]
 

Revision as of 16:59, 13 February 2025

Problem

In $\triangle ABC$ points $D$ and $E$ lie on $\overline{AB}$ so that $AD < AE < AB$, while points $F$ and $G$ lie on $\overline{AC}$ so that $AF < AG < AC$. Suppose $AD = 4$, $DE = 16$, $EB = 8$, $AF = 13$, $FG = 52$, and $GC = 26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G$ through $E$. The area of quadrilateral $DEGF$ is $288$. Find the area of heptagon $AFNBCEM$, as shown in the figure below.

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