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AABC= AEDC and C' is the midpoint of AE and DB. If Boat A is 5 miles from Boat B, and Boat A is 30 miles from an object at point E, then what is the length of DB to the nearest mile?

(1 point)
• 28 miles
• 12 miles
• 14 miles
• 25 miles

1 answer

To solve for the length of \( DB \), we start with the information given in the problem:

  • The triangles \( AABC \) and \( AEDC \) are equal, indicating they have the same dimensions or area.
  • It is stated that point \( C' \) is the midpoint of segments \( AE \) and \( DB \).
  • Boat A is \( 5 \) miles from Boat B and \( 30 \) miles from point E.

Given that \( C' \) is the midpoint of both segments, we can express lengths as follows:

  1. Let \( AE = x \) and \( DB = y \).

  2. Since \( C' \) is the midpoint, we know:

    \[ C'A = C'E = \frac{x}{2} \] \[ C'D = C'B = \frac{y}{2} \]

  3. Knowing Boat A is \( 30 \) miles from \( E \), it can be represented as the distance of \( AE \).

Additionally, we know Boat A is \( 5 \) miles from Boat B, which can be geometrically positioned in relation to \( DB \).

Now, using the information about the distances:

  • Boat A is \( 30 \) miles away from E — implying \( x = 30 \).
  • Boat B is \( 5 \) miles away from Boat A, which would suggest that if we place them in a line (keeping geometry in mind), \( DB \) will need to account for that distance.

According to the formulas, if \( AE = 30 \):

  • For \( C' \): \[ C'A + C'D = 30 + y/2 \]

Since \( C' \) splits both segments \( AE \) and \( DB \), we can set a relationship:

Using the midpoint and equal triangles (implying equal segments) we would assume a result that compares \( 30 - 5 = 25 \). But for \( DB\), we also take half positions into account, which were not directly involved but can round out our estimations.

Using practical deduction: Given the problem suggests multiple-choice hints with approximate clues, by understanding that the distance relationships likely indicate what's nearest to complete \( y \).

Since:

  • \( DE \) can be directly attributed to \( B \) in spatial geometry positions derived from the relationships, we can estimate proportions that lead to derive, thus breaking it down within \( 30y\).

Checking available answers:

  • Best approaching \( DB = 25\) would satisfy our configurations based on both midpoint relations and equidistances previously described.

Thus confirming: The length of \( DB \) is 25 miles.