2006 AIME I Problems/Problem 7
Problem
An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region to the area of shaded region
is 11/5. Find the ratio of shaded region
to the area of shaded region
.
Solution 1
Note that the apex of the angle is not on the parallel lines. Set up a coordinate proof.
Let the set of parallel lines be perpendicular to the x-axis, such that they cross it at . The base of region
is on the line
. The bigger base of region
is on the line
.
Let the top side of the angle be
and the bottom side be x-axis, as dividing the angle doesn't change the problem.
Since the area of the triangle is equal to ,
Region /Region
= \frac{11}{5}
= \frac{\frac 12(5-s)^2 - \frac 12(4-s)^2}{\frac 12(3-s)^2 - \frac12(2-s)^2}
$$ (Error compiling LaTeX. Unknown error_msg)
Solve this to find that .
Using the same reasoning as above, we get Region /Region
= \frac{\frac 12(7-s)^2 - \frac 12(6-s)^2}{\frac 12(1-s)^2}
\boxed{408}$.
== Solution 2 ==
Note that the sections between the two transversals can be divided into one small triangle and a number of trapezoids. Let one side length (not on a parallel line) of the small triangle be$ (Error compiling LaTeX. Unknown error_msg)xx^2
x
1$.
Move on to the second-smallest triangle, formed by attaching this triangle with the next trapezoid. Parallel lines give us similar triangles, so we know the proportion of this triangle to the previous triangle is$ (Error compiling LaTeX. Unknown error_msg)(\frac{x+1}{x})^2(x+1)^2
2x+1
2x+3
2x+7
2x+11$.
We can now use the given condition that the ratio of C and B is$ (Error compiling LaTeX. Unknown error_msg)\frac{11}{5}\frac{11}{5} = \frac{2x+7}{2x+3}
x = \frac{1}{6}
\frac{2(\frac{1}{6} + 11)}{(\frac{1}{6})^2} = \boxed{408.}$
See also
2006 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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