Question

laura is visiting new york city and decides to go up the empire state building to get a good view of the city, including views of the statue of liberty and the world trade center. she overhears a tour guide saying that the empire state building is around 4 miles from the world trade center. her brochure map states that the world trade center is 2 miles from the statue of liberty. if the three landmarks create a triangle, between what two distances is she currently from the statue of liberty?

Explanation:

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"concepts_used": [
"Triangle Inequality Theorem"
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</pre_analysis>

<reasoning>

Identify the given side lengths

Using the Triangle Inequality Theorem knowledge point
\[

$$\begin{aligned} &a = 4 \text{ miles (Empire State Building to World Trade Center)} \\ &b = 2 \text{ miles (World Trade Center to Statue of Liberty)} \\ &x = \text{distance from Empire State Building to Statue of Liberty} \end{aligned}$$

\]

Apply the triangle inequality bounds

Using the Triangle Inequality Theorem knowledge point
\[

$$\begin{aligned} &|a - b| < x < a + b \\ &|4 - 2| < x < 4 + 2 \\ &2 < x < 6 \end{aligned}$$

\]
</reasoning>

<answer>
Between 2 miles and 6 miles
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Word Problem",
"knowledge_point": [
"Mathematics",
"Geometry",
"Triangle Inequality Theorem"
]
}
</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Identify the given side lengths

Using the Triangle Inequality Theorem knowledge point
\[

$$\begin{aligned} &a = 4 \text{ miles (Empire State Building to World Trade Center)} \\ &b = 2 \text{ miles (World Trade Center to Statue of Liberty)} \\ &x = \text{distance from Empire State Building to Statue of Liberty} \end{aligned}$$

\]

Apply the triangle inequality bounds

Using the Triangle Inequality Theorem knowledge point
\[

$$\begin{aligned} &|a - b| < x < a + b \\ &|4 - 2| < x < 4 + 2 \\ &2 < x < 6 \end{aligned}$$

\]
</reasoning>

<answer>
Between 2 miles and 6 miles
</answer>

<post_analysis>
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"question_type": "Word Problem",
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"Mathematics",
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]
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