Question

a coach watches two swimmers, corey and stephen, while they practice. as corey swims directly past him, he thinks that stephen is a long way behind corey, but its hard to tell from where he is standing.

if the coach is 40 feet from corey and 50 feet from stephen, what range of distances can stephen and corey be apart?

between feet and feet.

Explanation:

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<reasoning>

Identify the given side lengths

Using the Triangle Inequality Theorem knowledge point
\[

$$\begin{aligned} &a = 40 \text{ feet (distance from coach to Corey)} \\ &b = 50 \text{ feet (distance from coach to Stephen)} \\ &x = \text{distance between Corey and Stephen} \end{aligned}$$

\]

Apply the triangle inequality bounds

Using the Triangle Inequality Theorem knowledge point
\[

$$\begin{aligned} &|a - b| < x < a + b \\ &|40 - 50| < x < 40 + 50 \\ &10 < x < 90 \end{aligned}$$

\]
</reasoning>

<answer>
If the coach is 40 feet from Corey and 50 feet from Stephen, what range of distances can Stephen and Corey be apart?

Between <blank>10</blank> feet and <blank>90</blank> feet.
</answer>

<plot>
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{
"type": "point",
"params": [[0, 0]],
"properties": {
"name": "Coach",
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"withLabel": true
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"name": "Corey",
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"type": "point",
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"name": "Stephen",
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]
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Answer:

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"concepts_used": [
"Triangle Inequality Theorem"
],
"new_concepts": [],
"current_concepts": [
"Triangle Inequality Theorem"
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}
</pre_analysis>

<reasoning>

Identify the given side lengths

Using the Triangle Inequality Theorem knowledge point
\[

$$\begin{aligned} &a = 40 \text{ feet (distance from coach to Corey)} \\ &b = 50 \text{ feet (distance from coach to Stephen)} \\ &x = \text{distance between Corey and Stephen} \end{aligned}$$

\]

Apply the triangle inequality bounds

Using the Triangle Inequality Theorem knowledge point
\[

$$\begin{aligned} &|a - b| < x < a + b \\ &|40 - 50| < x < 40 + 50 \\ &10 < x < 90 \end{aligned}$$

\]
</reasoning>

<answer>
If the coach is 40 feet from Corey and 50 feet from Stephen, what range of distances can Stephen and Corey be apart?

Between <blank>10</blank> feet and <blank>90</blank> feet.
</answer>

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"name": "Corey",
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"name": "Stephen",
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</plot>

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