1996 AHSME Problems/Problem 17

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), plain.E, fontsize(10)); label("$x$", (9,3.5), E, fontsize(10)); label("$x-2$", (0,5), plain.E, fontsize(10)); label("$y$", (5,7), N, fontsize(10)); label("$6$", (7.5,0), S, fontsize(10));[/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 $\boxed{\text{E}}$ . (The actual area is actually greater, since $\sqrt{3} > 1.7$ ).

1996 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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1996 AHSME Problems/Problem 17

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