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Superb Luigio must find the length, in feet, across the lake sketched above in order to swim to the other side and save Princess Mushroom from impending doom. He measured lengths represented by AB, EB, BD, and CD and determined them to be 2000 feet, 1792 feet, 256 feet, and 315 feet, respectively. Segments AC and DE intersect at B, and angles AEB and CDB have the same measure. What is the length of AE?

(A) 1920 feet

(B) 2015 feet

(C) 2110 feet

(D) 2205 feet

This problem can be solved using the concept of similar triangles because the angles \( \angle AEB \) and \( \angle CDB \) are given to be congruent. Therefore, triangles \( \triangle AEB \) and \( \triangle CDB \) are similar by the Angle-Angle (AA) similarity criterion.

Since the triangles are similar, the corresponding sides are proportional. Thus, we can set up the following proportion between the corresponding sides:

\[
\frac{AE}{EB} = \frac{CD}{BD}
\]

We are given:
- \( EB = 1792 \) feet
- \( BD = 256 \) feet
- \( CD = 315 \) feet

Substitute these values into the proportion:

\[
\frac{AE}{1792} = \frac{315}{256}
\]

Now, solve for \( AE \):

\[
AE = \frac{315}{256} \times 1792
\]

First, calculate the fraction \( \frac{315}{256} \):

\[
\frac{315}{256} \approx 1.2305
\]

Now multiply this by \( 1792 \):

\[
AE \approx 1.2305 \times 1792 = 2205.056 \text{ feet}
\]

Thus, the length of \( AE \) is approximately \( 2205 \) feet.

The correct answer is \( \boxed{2205} \) feet.

Would you like more details or have any questions about this solution? Here are some related questions to explore further:

1. Why are triangles \( AEB \) and \( CDB \) considered similar?
2. How does the property of proportional sides in similar triangles help solve the problem?
3. Can you generalize this method to other types of geometry problems involving lakes or distances?
4. What other geometric methods could be used to find the length across a lake?
5. What are other real-world applications where similar triangles are useful?

**Tip:** Always check for angle congruence when you suspect that two triangles might be similar—it's a key step in setting up proportions for solving problems!

Superb Luigio must find the length, in feet, across the lake sketched above in order to swim to the other side and save Princess Mushroom from impending doom. He measured lengths represented by AB, EB, BD, and CD and determined them to be 2000 feet, 1792 feet, 256 feet, and 315 feet, respectively. Segments AC and DE intersect at B, and angles AEB and CDB have the same measure. What is the length of AE? (A) 1920 feet (B) 2015 feet (C) 2110 feet (D) 2205 feet

Solve a geometry problem involving similar triangles and proportions. Given angles AEB and CDB are congruent, determine the length of AE using the sides of two triangles. Learn the step-by-step solution and apply the AA similarity theorem to find the answer.

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