ABC = EDC and C is the midpoint of AE and DB. If Boat A is 9 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 answer

Given the problem, we are given that \( ABC = EDC \) and that point \( C \) is the midpoint of both segments \( AE \) and \( DB \).

Let's define the distances as follows:

  • Let the distance from boat A to point E (denote it as \( A \) to \( E \)) be 30 miles, so \( AE = 30 \) miles.
  • Let the distance between boat A and boat B be \( AB = 9 \) miles.

Since \( C \) is the midpoint of \( AE \), we can find the distance from \( A \) to \( C \): \[ AC = \frac{AE}{2} = \frac{30}{2} = 15 \text{ miles} \]

Now, let's consider the configuration of the points. Given that \( AB = 9 \) miles, we recognize that the distance from boat B to point C is given by: \[ BC = AB + AC = 9 + 15 = 24 \text{ miles} \]

Because \( C \) is also the midpoint of \( DB \), we can denote: \[ DC = \frac{DB}{2} \]

Since \( ABC = EDC \) signifies that triangles \( ABC \) and \( EDC \) are congruent, we know that \( AC = EC \) and \( BC = DC \).

Now we already calculated \( AC = 15 \) miles, so: \[ EC = AC = 15 \text{ miles} \]

Using the information that \( BC = DC \) and we already found \( BC = 24 \) miles: \[ DC = 24 \text{ miles} \]

Now we can find \( DB \) using the relationship \( DB = 2 \times DC \): \[ DB = 2 \times 24 = 48 \text{ miles} \]

Thus, the length of \( DB \) to the nearest mile is \( \boxed{48} \).