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Difference between revisions of "2025 AIME I Problems/Problem 2"

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==Problem==
 
==Problem==
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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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On <math>\triangle ABC</math> points <math>A</math>, <math>D</math>, <math>E</math>, and <math>B</math> lie in that order on side <math>\overline{AB}</math> with <math>AD = 4</math>, <math>DE = 16</math>, and <math>EB = 8</math>. Points <math>A</math>, <math>F</math>, <math>G</math>, and <math>C</math> lie in that order on side <math>\overline{AC}</math> with <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>. Quadrilateral <math>DEGF</math> has area <math>288</math>. Find the area of heptagon <math>AFNBCEM</math>.
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<asy>
 
<asy>
 
unitsize(14);
 
unitsize(14);
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Because of reflections, and various triangles having the same bases, we can conclude that <math>|AFNBCEM| = |ABC|</math>. Through the given lengths of <math>4-16-8</math> on the left and <math>13-52-26</math> on the right, we conclude that the lines through <math>\triangle ABC</math> are parallel, and the sides are in a <math>1:4:2</math> ratio. Because these lines are parallel, we can see that <math>ADF,~AEG,~ABC</math>, are similar, and from our earlier ratio, we can give the triangles side ratios of <math>1:5:7</math>, or area ratios of <math>1:25:49</math>. Quadrilateral <math>DEGF</math> corresponds to the <math>|AEG|-|ADF|</math>, which corresponds to the ratio <math>25-1=24</math>. Dividing <math>288</math> by <math>24</math>, we get <math>12</math>, and finally multiplying <math>12 \cdot 49</math> gives us our answer of <math>\boxed{588}</math>
 
Because of reflections, and various triangles having the same bases, we can conclude that <math>|AFNBCEM| = |ABC|</math>. Through the given lengths of <math>4-16-8</math> on the left and <math>13-52-26</math> on the right, we conclude that the lines through <math>\triangle ABC</math> are parallel, and the sides are in a <math>1:4:2</math> ratio. Because these lines are parallel, we can see that <math>ADF,~AEG,~ABC</math>, are similar, and from our earlier ratio, we can give the triangles side ratios of <math>1:5:7</math>, or area ratios of <math>1:25:49</math>. Quadrilateral <math>DEGF</math> corresponds to the <math>|AEG|-|ADF|</math>, which corresponds to the ratio <math>25-1=24</math>. Dividing <math>288</math> by <math>24</math>, we get <math>12</math>, and finally multiplying <math>12 \cdot 49</math> gives us our answer of <math>\boxed{588}</math>
  
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~shreyan.chethan
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~shreyan.chethan, cleaned up by cweu001
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==Solution 3==
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By area lemma, we can see that the areas of the shaded areas are equivalent to the areas of the unshaded areas. Thus, we see that the desired area is equivalent to the area of the triangle <math>\triangle ABC</math>. Since <math>AF : FG : GC = 1 : 4 : 2</math>, we have <math>[ADF]:[AEG]:[ABC] = 1:25:49</math>, meaning <math>[ADF]:[DEGF]:[BEGC] = 1:24:24</math>. Thus, since <math>\frac{[DEGF]}{ABC} = \frac{24}{49}</math>, we can calculate <math>[ABC] = 588</math>.
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 +
~cweu001, cleaned up by shreyan.chethan
  
 
==Video Solution 1 by SpreadTheMathLove==
 
==Video Solution 1 by SpreadTheMathLove==
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~MC
 
~MC
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==Video Solution by Mathletes Corner==
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https://www.youtube.com/watch?v=fVBk2vOusio&t=3s
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 +
~GP102
  
 
==See also==
 
==See also==

Latest revision as of 11:01, 13 May 2025

Problem

On $\triangle ABC$ points $A$, $D$, $E$, and $B$ lie in that order on side $\overline{AB}$ with $AD = 4$, $DE = 16$, and $EB = 8$. Points $A$, $F$, $G$, and $C$ lie in that order on side $\overline{AC}$ with $AF = 13$, $FG = 52$, and $GC = 26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G$ through $E$. Quadrilateral $DEGF$ has area $288$. Find the area of heptagon $AFNBCEM$.

[asy] unitsize(14); 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; filldraw(A--F--N--B--C--E--M--cycle, lightgray); draw(A--B--C--cycle); draw(D--M); draw(N--G); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(M); dot(N); label("$A$", A, dir(90)); label("$B$", B, dir(225)); label("$C$", C, dir(315)); label("$D$", D, dir(135)); label("$E$", E, dir(135)); label("$F$", F, dir(45)); label("$G$", G, dir(45)); label("$M$", M, dir(45)); label("$N$", N, dir(135)); [/asy]

Solution 1

Note that the triangles outside $\triangle ABC$ have the same height as the unshaded triangles in $\triangle ABC$. Since they have the same bases, the area of the heptagon is the same as the area of triangle $ABC$. Therefore, we need to calculate the area of $\triangle ABC$. Denote the length of $DF$ as $x$ and the altitude of $A$ to $DF$ as $h$. Since $\triangle ADF \sim \triangle AEG$, $EG = 5x$ and the altitude of $DFGE$ is $4h$. The area $[DFGE] = \frac{5x + x}{2} \cdot 4h = 3x \cdot 4h = 12xh = 288 \implies xh = 24$. The area of $\triangle ABC$ is equal to $\frac{1}{2} 7x \cdot 7h = \frac{1}{2} 49xh = \frac{1}{2} 49 \cdot 24 = \frac{1}{2} 1176 = \boxed{588}$.

~alwaysgonnagiveyouup

Solution 2

Because of reflections, and various triangles having the same bases, we can conclude that $|AFNBCEM| = |ABC|$. Through the given lengths of $4-16-8$ on the left and $13-52-26$ on the right, we conclude that the lines through $\triangle ABC$ are parallel, and the sides are in a $1:4:2$ ratio. Because these lines are parallel, we can see that $ADF,~AEG,~ABC$, are similar, and from our earlier ratio, we can give the triangles side ratios of $1:5:7$, or area ratios of $1:25:49$. Quadrilateral $DEGF$ corresponds to the $|AEG|-|ADF|$, which corresponds to the ratio $25-1=24$. Dividing $288$ by $24$, we get $12$, and finally multiplying $12 \cdot 49$ gives us our answer of $\boxed{588}$

~shreyan.chethan, cleaned up by cweu001

Solution 3

By area lemma, we can see that the areas of the shaded areas are equivalent to the areas of the unshaded areas. Thus, we see that the desired area is equivalent to the area of the triangle $\triangle ABC$. Since $AF : FG : GC = 1 : 4 : 2$, we have $[ADF]:[AEG]:[ABC] = 1:25:49$, meaning $[ADF]:[DEGF]:[BEGC] = 1:24:24$. Thus, since $\frac{[DEGF]}{ABC} = \frac{24}{49}$, we can calculate $[ABC] = 588$.

~cweu001, cleaned up by shreyan.chethan

Video Solution 1 by SpreadTheMathLove

https://www.youtube.com/watch?v=J-0BapU4Yuk

Video Solution(Fast! Easy!)

https://youtu.be/LQyncubz30U

~MC

Video Solution by Mathletes Corner

https://www.youtube.com/watch?v=fVBk2vOusio&t=3s

~GP102

See also

2025 AIME I (Problems • Answer Key • Resources)
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Problem 1 		Followed by
Problem 3
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All AIME Problems and Solutions

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