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 $\mathcal{C}$ 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

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 $x = 7$ . Cutting the angle by half don't change the problem. After the cut, the top side of the angle will be $y = x - s$ ; the bottom side will be x-axis.

Since the area of the triangle is equal to $\frac{1}{2}bh$ ,

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

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

By a similar method, $\frac{Region \mathcal{D}}{Region \mathcal{A}} = \frac{\frac 12(7-s)^2 - \frac 12(6-s)^2}{\frac 12(1-s)^2}$ is $408$ .

2006 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
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2006 AIME I Problems/Problem 7

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