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Prompt: In a regular hexagon $ABCDEF$, let $G$ be the intersection of $\overline{AD}$ and $\overline{CE}$, and let $H$ be the intersection of $\overline{BD}$ and $\overline{AF}$. If the side length of the hexagon is $1$, find the length of $\overline{GH}$.

To solve this problem, we can start by drawing a diagram of the regular hexagon $ABCDEF$ and labeling the given points $G$ and $H$.



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Since the hexagon is regular, all interior angles are congruent and measure $120^\circ$. This means that $\triangle ACG$ and $\triangle BFG$ are both $30^\circ-60^\circ-90^\circ$ triangles.



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In a $30^\circ-60^\circ-90^\circ$ triangle, the side lengths are in the ratio $1:\sqrt{3}:2$. Therefore, we can find the length of $\overline{AG}$ by applying this ratio to $\triangle ACG$: $$\overline{AG} = \frac{\sqrt{3}}{2}\cdot \overline{AC} = \frac{\sqrt{3}}{2}\cdot 1 = \frac{\sqrt{3}}{2}.$$



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Similarly, we can find the length of $\overline{BH}$ by applying the ratio to $\triangle BFG$: $$\overline{BH} = \frac{\sqrt{3}}{2}\cdot \overline{BF} = \frac{\sqrt{3}}{2}\cdot 1 = \frac{\sqrt{3}}{2}.$$



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Now, let's examine $\triangle ABH$ and $\triangle AGH$. We have that $\angle ABH = 120^\circ$ and $\angle AGH = 60^\circ$, so these triangles are also $30^\circ-60^\circ-90^\circ$ triangles.



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By the properties of these triangles, we can determine that $\overline{GH}$ is the shorter leg of $\triangle AGH$. Therefore, we can find the length of $\overline{GH}$ by multiplying the length of $\overline{AG}$ by $\frac{1}{2}$ or by multiplying the length of $\overline{AH}$ by $\frac{\sqrt{3}}{2}$.



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However, we do not know the length of $\overline{AH}$. Instead, we can use the fact that $\overline{AB} = 1$ and that $\overline{BH} = \frac{\sqrt{3}}{2}$. By applying the Pythagorean theorem to $\triangle ABH$, we can find the length of $\overline{AH}$: $$\overline{AH}^2 + \overline{BH}^2 = \overline{AB}^2$$ $$\overline{AH}^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = 1^2$$ $$\overline{AH}^2 = 1 - \frac{3}{4}$$ $$\overline{AH} = \frac{1}{2}$$



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Now, we can find the length of $\overline{GH}$ by multiplying the length of $\overline{AH}$ by $\frac{\sqrt{3}}{2}$ or by finding the difference between $\overline{AG}$ and $\overline{AH}$: $$\overline{GH} = \overline{AG} - \overline{AH} = \frac{\sqrt{3}}{2} - \frac{1}{2}$$

However



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the most efficient method is to find $\overline{GH}$ is to notice that $\triangle ABH$ and $\triangle AGH$ are similar. Then $\overline{GH} = \frac{1}{2} \cdot \overline{AB} = \frac{1}{2} \cdot 1 = \frac{1}{2}$.



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Correct - LLM is appropriate


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Correct - Python is appropriate