Difference between revisions of "1996 AHSME Problems/Problem 17"

(Solution)
Line 44: Line 44:
 
The area of the square is thus <math>xy = 6\sqrt{3}(18 - 2\sqrt{3}) = 108\sqrt{3} - 36</math>.
 
The area of the square is thus <math>xy = 6\sqrt{3}(18 - 2\sqrt{3}) = 108\sqrt{3} - 36</math>.
  
Using the approximation <math>\sqrt{3} \approx 1.7</math>, we get an area of just under <math>147.6</math>, which is closest to answer <math>\boxed{D}</math>.  (The actual area is actually greater, since <math>\sqrt{3} > 1.7</math>).
+
Using the approximation <math>\sqrt{3} \approx 1.7</math>, we get an area of just under <math>147.6</math>, which is closest to answer <math>\boxed{E}</math>.  (The actual area is actually greater, since <math>\sqrt{3} > 1.7</math>).
  
 
==See also==
 
==See also==
 
{{AHSME box|year=1996|num-b=16|num-a=18}}
 
{{AHSME box|year=1996|num-b=16|num-a=18}}

Revision as of 15:29, 19 August 2011

Problem

In rectangle $ABCD$, angle $C$ is trisected by $\overline{CF}$ and $\overline{CE}$, where $E$ is on $\overline{AB}$, $F$ is on $\overline{AD}$, $BE=6$ and $AF=2$. Which of the following is closest to the area of the rectangle $ABCD$? [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] $\text{(A)}\ 110\qquad\text{(B)}\ 120\qquad\text{(C)}\ 130\qquad\text{(D)}\ 140\qquad\text{(E)}\ 150$

Solution

Since $\angle C = 90^\circ$, each of the three smaller angles is $30^\circ$, and $\triangle BEC$ and $\triangle CDF$ are both $30-60-90$ 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]

Defining the variables as illustrated above, we have $x = 6\sqrt{3}$ from $\triangle BEC$

Then $x-2 = 6\sqrt{3} - 2$, and $y = \sqrt{3} (6 \sqrt{3} - 2) = 18 - 2\sqrt{3}$.

The area of the square is thus $xy = 6\sqrt{3}(18 - 2\sqrt{3}) = 108\sqrt{3} - 36$.

Using the approximation $\sqrt{3} \approx 1.7$, we get an area of just under $147.6$, which is closest to answer $\boxed{E}$. (The actual area is actually greater, since $\sqrt{3} > 1.7$).

See also

1996 AHSME (ProblemsAnswer KeyResources)
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
Invalid username
Login to AoPS