1994 AJHSME Problems/Problem 12

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Problem 12

Each of the three large squares shown below is the same size. Segments that intersect the sides of the squares intersect at the midpoints of the sides. How do the shaded areas of these squares compare?

[asy] unitsize(36); fill((0,0)--(1,0)--(1,1)--cycle,gray); fill((1,1)--(1,2)--(2,2)--cycle,gray); draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); draw((1,0)--(1,2)); draw((0,0)--(2,2));  fill((3,1)--(4,1)--(4,2)--(3,2)--cycle,gray); draw((3,0)--(5,0)--(5,2)--(3,2)--cycle); draw((4,0)--(4,2)); draw((3,1)--(5,1));  fill((6,1)--(6.5,0.5)--(7,1)--(7.5,0.5)--(8,1)--(7.5,1.5)--(7,1)--(6.5,1.5)--cycle,gray); draw((6,0)--(8,0)--(8,2)--(6,2)--cycle); draw((6,0)--(8,2)); draw((6,2)--(8,0)); draw((7,0)--(6,1)--(7,2)--(8,1)--cycle);  label("$I$",(1,2),N); label("$II$",(4,2),N); label("$III$",(7,2),N); [/asy]

$\text{(A)}\ \text{The shaded areas in all three are equal.}$

$\text{(B)}\ \text{Only the shaded areas of }I\text{ and }II\text{ are equal.}$

$\text{(C)}\ \text{Only the shaded areas of }I\text{ and }III\text{ are equal.}$

$\text{(D)}\ \text{Only the shaded areas of }II\text{ and }III\text{ are equal.}$

$\text{(E)}\ \text{The shaded areas of }I, II\text{ and }III\text{ are all different.}$

Solution

Square II clearly has $1/4$ shaded. Partitioning square I into eight right triangles also shows $1/4$ of it is shaded. Lastly, square III can be partitioned into sixteen triangles, and because four are shaded, $1/4$ of the total square is shaded. $\boxed{\text{(A)}}$.

See Also

1994 AJHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
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All AJHSME/AMC 8 Problems and Solutions
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