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2006 AIME I Problems/Problem 7: Difference between revisions

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== Solution ==
== Solution 1 ==
Note that the apex of the angle is not on the parallel lines. Set up a [[coordinate proof]].
Note that the apex of the angle is not on the parallel lines. Set up a [[coordinate proof]].


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Using the same reasoning as above, we get <math>\frac{\textrm{Region\ }\mathcal{D}}{\textrm{Region\ }\mathcal{A}} = \frac{\frac 12(7-s)^2 - \frac 12(6-s)^2}{\frac 12(1-s)^2}</math>, which is <math>\boxed{408}</math>.
Using the same reasoning as above, we get <math>\frac{\textrm{Region\ }\mathcal{D}}{\textrm{Region\ }\mathcal{A}} = \frac{\frac 12(7-s)^2 - \frac 12(6-s)^2}{\frac 12(1-s)^2}</math>, which is <math>\boxed{408}</math>.
== 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 <math>x</math> and the area of it be <math>x^2</math>. Also, let all sections of the line on the same side as the side with length <math>x</math> on a trapezoid be equal to <math>1</math>.
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 <math>(\frac{x+1}{x})^2</math>. Multiplying, we get <math>(x+1)^2</math> as the area of the triangle, so the area of the trapezoid is <math>2x+1</math>. Repeating this process, we get that the area of B is <math>2x+3</math>, the area of C is <math>2x+7</math>, and the area of D is <math>2x+11</math>.
We can now use the given condition that the ratio of C and B is <math>\frac{11}{5}</math>.
<math>\frac{11}{5} = \frac{2x+7}{2x+3}</math> gives us <math>x = \frac{1}{6}</math>
So now we compute the ratio of D and A, which is <math>\frac{2(\frac{1}{6} + 11)}{(\frac{1}{6})^2} = \boxed{408.}</math>


== See also ==
== See also ==

Revision as of 21:00, 13 January 2016

Problem

An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region $\mathcal{C}$ to the area of shaded region $\mathcal{B}$ is 11/5. Find the ratio of shaded region $\mathcal{D}$ to the area of shaded region $\mathcal{A}.$

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 $0, 1, 2 \ldots$. The base of region $\mathcal{A}$ is on the line $x = 1$. The bigger base of region $\mathcal{D}$ is on the line $x = 7$. Let the top side of the angle be $y = x - s$ 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 $\frac{1}{2}bh$,

\[\frac{\textrm{Region\ }\mathcal{C}}{\textrm{Region\ }\mathcal{B}} = \frac{11}{5} = \frac{\frac 12(5-s)^2 - \frac 12(4-s)^2}{\frac 12(3-s)^2 - \frac12(2-s)^2}\]

Solve this to find that $s = \frac{5}{6}$.

Using the same reasoning as above, we get $\frac{\textrm{Region\ }\mathcal{D}}{\textrm{Region\ }\mathcal{A}} = \frac{\frac 12(7-s)^2 - \frac 12(6-s)^2}{\frac 12(1-s)^2}$, which is $\boxed{408}$.

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 $x$ and the area of it be $x^2$. Also, let all sections of the line on the same side as the side with length $x$ on a trapezoid be equal to $1$.

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 $(\frac{x+1}{x})^2$. Multiplying, we get $(x+1)^2$ as the area of the triangle, so the area of the trapezoid is $2x+1$. Repeating this process, we get that the area of B is $2x+3$, the area of C is $2x+7$, and the area of D is $2x+11$.

We can now use the given condition that the ratio of C and B is $\frac{11}{5}$.

$\frac{11}{5} = \frac{2x+7}{2x+3}$ gives us $x = \frac{1}{6}$

So now we compute the ratio of D and A, which is $\frac{2(\frac{1}{6} + 11)}{(\frac{1}{6})^2} = \boxed{408.}$

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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