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Difference between revisions of "2024 AIME II Problems/Problem 5"

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Let ABCDEF be a convex equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are extensions of segments AB, CD, and EF has side lengths 200, 240, and 300. Find the side length of the hexagon.
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==Solution 1==
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(Sorry i have zero idea how to make drawings)
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Let <math>AB \cap DC</math>, <math>CD \cap FE</math>, and <math>BA \cap EF</math> be P, Q, and R, respectively. Let <math>QR=200, RP=300, and PQ=240</math>. Notice that all smaller triangles formed are all similar to the larger <math>(200,240,300)</math> triangle. Let the side length of the hexagon be x. Triangle <math> \triangle BCP \sim \triangle RQP</math>, so <math>\frac{BC}{BP} =\frac{x}{BP} =\frac{200}{300} \implies BP=\frac{3x}{2}</math>. Triangle <math>\triangle AFR \sim \triangle PQR</math>, so <math>\frac{AF}{AR}=\frac{x}{AR} = \frac{240}{300} \implies AR=\frac{5x}{4}</math>. We know <math>RA+AB+BP=300</math>, so <math>\frac{5}{4}x + x + \frac{3}{2}x = 300</math>. Solving, we get <math>x=\boxed{080}</math>.

Revision as of 21:12, 8 February 2024

Let ABCDEF be a convex equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are extensions of segments AB, CD, and EF has side lengths 200, 240, and 300. Find the side length of the hexagon.

Solution 1

(Sorry i have zero idea how to make drawings) Let $AB \cap DC$, $CD \cap FE$, and $BA \cap EF$ be P, Q, and R, respectively. Let $QR=200, RP=300, and PQ=240$. Notice that all smaller triangles formed are all similar to the larger $(200,240,300)$ triangle. Let the side length of the hexagon be x. Triangle $\triangle BCP \sim \triangle RQP$, so $\frac{BC}{BP} =\frac{x}{BP} =\frac{200}{300} \implies BP=\frac{3x}{2}$. Triangle $\triangle AFR \sim \triangle PQR$, so $\frac{AF}{AR}=\frac{x}{AR} = \frac{240}{300} \implies AR=\frac{5x}{4}$. We know $RA+AB+BP=300$, so $\frac{5}{4}x + x + \frac{3}{2}x = 300$. Solving, we get $x=\boxed{080}$.

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