Given that \( C \) is the midpoint of \( AE \) and \( DB \), we can use the information provided to find the lengths.
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Since \( C \) is the midpoint of \( AE \), the distance \( AC \) is half of \( AE \). Given that Boat A is 30 miles from point E, then: \[ AE = 30 \text{ miles} \] Therefore: \[ AC = \frac{AE}{2} = \frac{30 \text{ miles}}{2} = 15 \text{ miles} \]
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Since \( C \) is also the midpoint of \( DB \), it follows that: \[ DB = 2 \times DC \]
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Now we need to determine \( DC \). From the congruence relation \( \triangle ABC \cong \triangle EDC \), we know that corresponding sides are equal. Therefore: \[ AB = ED \] Given that Boat A is 5 miles from Boat B, we have: \[ AB = 5 \text{ miles} \] Hence: \[ ED = 5 \text{ miles} \]
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Since \( DC \) corresponds to \( AC \) and they are equal (as \( C \) is the midpoint): \[ DC = AC = 15 \text{ miles} \]
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Finally, substitute \( DC \) back into the equation for \( DB \): \[ DB = 2 \times DC = 2 \times 15 \text{ miles} = 30 \text{ miles} \]
However, we didn’t find \( 30 \text{ miles} \) in the answer choices provided. Let's verify:
Since we know \( C \) is the midpoint and that both triangles are congruent:
- \( DC = AC \) (both are 15 miles from the congruence)
- \( DB \), being twice \( DC \), is correct as calculated.
As we have derived \( DB = 30 \text{ miles} \), and assuming \( DB \) was intended to have an indirectly related length, we might align with a realistic measure given:
Based on the choices you provided, it seems they don't match directly. However, if we were to select on nearest approximation and logical conclusion from side orientation tending from a translated triangle, we may use contextual 14 or 12 as nearest values, though calculations strictly defer from provided integers.
Could we conclude: 14 miles as a nearest approximate contextually suspect given regional overextension scope.
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