Difference between revisions of "2021 AIME II Problems/Problem 2"
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==Problem== | ==Problem== | ||
Equilateral triangle <math>ABC</math> has side length <math>840</math>. Point <math>D</math> lies on the same side of line <math>BC</math> as <math>A</math> such that <math>\overline{BD} \perp \overline{BC}</math>. The line <math>\ell</math> through <math>D</math> parallel to line <math>BC</math> intersects sides <math>\overline{AB}</math> and <math>\overline{AC}</math> at points <math>E</math> and <math>F</math>, respectively. Point <math>G</math> lies on <math>\ell</math> such that <math>F</math> is between <math>E</math> and <math>G</math>, <math>\triangle AFG</math> is isosceles, and the ratio of the area of <math>\triangle AFG</math> to the area of <math>\triangle BED</math> is <math>8:9</math>. Find <math>AF</math>. | Equilateral triangle <math>ABC</math> has side length <math>840</math>. Point <math>D</math> lies on the same side of line <math>BC</math> as <math>A</math> such that <math>\overline{BD} \perp \overline{BC}</math>. The line <math>\ell</math> through <math>D</math> parallel to line <math>BC</math> intersects sides <math>\overline{AB}</math> and <math>\overline{AC}</math> at points <math>E</math> and <math>F</math>, respectively. Point <math>G</math> lies on <math>\ell</math> such that <math>F</math> is between <math>E</math> and <math>G</math>, <math>\triangle AFG</math> is isosceles, and the ratio of the area of <math>\triangle AFG</math> to the area of <math>\triangle BED</math> is <math>8:9</math>. Find <math>AF</math>. | ||
+ | |||
+ | |||
+ | <asy> | ||
+ | pair A,B,C,D,E,F,G; | ||
+ | B=origin; | ||
+ | A=5*dir(60); | ||
+ | C=(5,0); | ||
+ | E=0.6*A+0.4*B; | ||
+ | F=0.6*A+0.4*C; | ||
+ | G=rotate(240,F)*A; | ||
+ | D=extension(E,F,B,dir(90)); | ||
+ | draw(D--G--A,grey); | ||
+ | draw(B--0.5*A+rotate(60,B)*A*0.5,grey); | ||
+ | draw(A--B--C--cycle,linewidth(1.5)); | ||
+ | dot(A^^B^^C^^D^^E^^F^^G); | ||
+ | label("$A$",A,dir(90)); | ||
+ | label("$B$",B,dir(225)); | ||
+ | label("$C$",C,dir(-45)); | ||
+ | label("$D$",D,dir(180)); | ||
+ | label("$E$",E,dir(-45)); | ||
+ | label("$F$",F,dir(225)); | ||
+ | label("$G$",G,dir(0)); | ||
+ | label("$\ell$",midpoint(E--F),dir(90)); | ||
+ | </asy> | ||
+ | |||
==Solution== | ==Solution== | ||
SOMEBODY DELETED A SOLUTION - PLEASE WAIT FOR THE ORIGINAL AUTHOR TO COME BACK ~ARCTICTURN | SOMEBODY DELETED A SOLUTION - PLEASE WAIT FOR THE ORIGINAL AUTHOR TO COME BACK ~ARCTICTURN |
Revision as of 19:46, 22 March 2021
Problem
Equilateral triangle has side length . Point lies on the same side of line as such that . The line through parallel to line intersects sides and at points and , respectively. Point lies on such that is between and , is isosceles, and the ratio of the area of to the area of is . Find .
Solution
SOMEBODY DELETED A SOLUTION - PLEASE WAIT FOR THE ORIGINAL AUTHOR TO COME BACK ~ARCTICTURN
See also
2021 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.