1989 AJHSME Problems/Problem 15

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Problem

The area of the shaded region $\text{BEDC}$ in parallelogram $\text{ABCD}$ is

[asy] unitsize(10); pair A,B,C,D,E; A=origin; B=(4,8); C=(14,8); D=(10,0); E=(4,0); draw(A--B--C--D--cycle); fill(B--E--D--C--cycle,gray); label("A",A,SW); label("B",B,NW); label("C",C,NE); label("D",D,SE); label("E",E,S); label("$10$",(9,8),N); label("$6$",(7,0),S); label("$8$",(4,4),W); draw((3,0)--(3,1)--(4,1)); [/asy]

$\text{(A)}\ 24 \qquad \text{(B)}\ 48 \qquad \text{(C)}\ 60 \qquad \text{(D)}\ 64 \qquad \text{(E)}\ 80$

Solution

Let $!ABC!$ denote the area of figure $ABC$.

Clearly, $!BEDC!=!ABCD!-!ABE!$. Using basic area formulas, \[!ABCD!=(BC)(BE)=80\] \[!ABE!=(BE)(AE)/2 = 4(AE)\]

Since $AE+EC=BC=10$ and $EC=6$, $AE=4$ and the area of $\triangle ABE$ is $4(4)=16$.

Finally, we have $!BEDC!=80-16=64\rightarrow \boxed{\text{D}}$

See Also

1989 AJHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
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
All AJHSME/AMC 8 Problems and Solutions