Difference between revisions of "2025 AIME I Problems/Problem 2"
Mathkiddus (talk | contribs) (→Problem) |
Mathkiddus (talk | contribs) (→Problem) |
||
| Line 1: | Line 1: | ||
==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. | ||
| + | <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> | ||
Revision as of 17:01, 13 February 2025
Problem
In 
points 
and 
lie on 
so that 
, while points 
and 
lie on 
so that 
. Suppose 
, 
, 
, 
, 
, and 
. Let 
be the reflection of through , and let 
be the reflection of through . The area of quadrilateral 
is 
. Find the area of heptagon 
, as shown in the figure below.
![[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]](http://latex.artofproblemsolving.com/6/4/b/64b0d84a95e0388e37622b54a251d836088fd97c.png)
