Difference between revisions of "2021 AIME I Problems/Problem 2"
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==Problem== | ==Problem== | ||
In the diagram below, <math>ABCD</math> is a rectangle with side lengths <math>AB=3</math> and <math>BC=11</math>, and <math>AECF</math> is a rectangle with side lengths <math>AF=7</math> and <math>FC=9,</math> as shown. The area of the shaded region common to the interiors of both rectangles is <math>\frac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | In the diagram below, <math>ABCD</math> is a rectangle with side lengths <math>AB=3</math> and <math>BC=11</math>, and <math>AECF</math> is a rectangle with side lengths <math>AF=7</math> and <math>FC=9,</math> as shown. The area of the shaded region common to the interiors of both rectangles is <math>\frac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
| + | <asy> | ||
| + | pair A, B, C, D, E, F; | ||
| + | A = (0,3); | ||
| + | B=(0,0); | ||
| + | C=(11,0); | ||
| + | D=(11,3); | ||
| + | E=foot(C, A, (9/4,0)); | ||
| + | F=foot(A, C, (35/4,3)); | ||
| + | draw(A--B--C--D--cycle); | ||
| + | draw(A--E--C--F--cycle); | ||
| + | filldraw(A--(9/4,0)--C--(35/4,3)--cycle,gray*0.5+0.5*lightgray); | ||
| + | dot(A^^B^^C^^D^^E^^F); | ||
| + | label("$A$", A, W); | ||
| + | label("$B$", B, W); | ||
| + | label("$C$", C, (1,0)); | ||
| + | label("$D$", D, (1,0)); | ||
| + | label("$F$", F, N); | ||
| + | label("$E$", E, S); | ||
| + | </asy> | ||
==Solution== | ==Solution== | ||
Revision as of 16:46, 11 March 2021
Problem
In the diagram below, 
is a rectangle with side lengths 
and 
, and 
is a rectangle with side lengths 
and 
as shown. The area of the shaded region common to the interiors of both rectangles is 
, where 
and 
are relatively prime positive integers. Find 
.
![[asy] pair A, B, C, D, E, F; A = (0,3); B=(0,0); C=(11,0); D=(11,3); E=foot(C, A, (9/4,0)); F=foot(A, C, (35/4,3)); draw(A--B--C--D--cycle); draw(A--E--C--F--cycle); filldraw(A--(9/4,0)--C--(35/4,3)--cycle,gray*0.5+0.5*lightgray); dot(A^^B^^C^^D^^E^^F); label("$A$", A, W); label("$B$", B, W); label("$C$", C, (1,0)); label("$D$", D, (1,0)); label("$F$", F, N); label("$E$", E, S); [/asy]](http://latex.artofproblemsolving.com/c/e/7/ce7ef019f55e9d0cf7364f8d93782a020489c947.png)
Solution
See also
| 2021 AIME I (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. 
