1996 AHSME Problems/Problem 17
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Problem
In rectangle 
, angle 
is trisected by 
and 
, where 
is on 
, 
is on 
, 
and 
. Which of the following is closest to the area of the rectangle ?
![[asy] pair A=origin, B=(10,0), C=(10,7), D=(0,7), E=(5,0), F=(0,2); draw(A--B--C--D--cycle, linewidth(0.8)); draw(E--C--F); dot(A^^B^^C^^D^^E^^F); label("$A$", A, dir((5, 3.5)--A)); label("$B$", B, dir((5, 3.5)--B)); label("$C$", C, dir((5, 3.5)--C)); label("$D$", D, dir((5, 3.5)--D)); label("$E$", E, dir((5, 3.5)--E)); label("$F$", F, dir((5, 3.5)--F)); label("$2$", (0,1), dir(0)); label("$6$", (7.5,0), N);[/asy]](http://latex.artofproblemsolving.com/6/5/1/6514d7d3f38ac6ef7e2a61a9c761d3ea53e8ef95.png)
Solution
Since 
, each of the three smaller angles is 
, and 
and 
are both 
triangles.
![[asy] pair A=origin, B=(10,0), C=(10,7), D=(0,7), E=(5,0), F=(0,2); draw(A--B--C--D--cycle, linewidth(0.8)); draw(E--C--F); dot(A^^B^^C^^D^^E^^F); label("$A$", A, dir((5, 3.5)--A)); label("$B$", B, dir((5, 3.5)--B)); label("$C$", C, dir((5, 3.5)--C)); label("$D$", D, dir((5, 3.5)--D)); label("$E$", E, dir((5, 3.5)--E)); label("$F$", F, dir((5, 3.5)--F)); label("$2$", (0,1), W); label("$x$", (9,3.5), E); label("$x-2$", (.2,5), W); label("$y$", (5,7), N); label("$6$", (7.5,0), S);[/asy]](http://latex.artofproblemsolving.com/8/6/d/86d4d83cf5e601d8af5a444df895d3fd60994513.png)
Defining the variables as illustrated above, we have 
from
Then 
, and 
.
The area of the square is thus 
.
Using the approximation 
, we get an area of just under 
, which is closest to answer 
. (The actual area is actually greater, since 
).
See also
| 1996 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 16 |
Followed by Problem 18 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
