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The image shows a set of similar triangles where we are tasked with determining the lengths of segments \( BG \) and \( CF \), labeled as \( X \) and \( Y \) respectively. These segments are part of a right triangle configuration, with side lengths given in the diagram.

We can use properties of similar triangles to set up proportions to solve for \( X \) and \( Y \). The triangles we are dealing with are:

1. \( \triangle ABD \)
2. \( \triangle ACD \)
3. \( \triangle AED \)

### Known Values:
- \( AG = 16 \, \text{m} \)
- \( GF = 20 \, \text{m} \)
- \( FE = 24 \, \text{m} \)
- \( DE = 45 \, \text{m} \)

### Steps to Solve:

#### 1. Set up a proportion for \( X = BG \):

We use similar triangles \( \triangle ABD \) and \( \triangle AED \). The sides corresponding to \( BG \) and \( AG \) relate proportionally to the overall sides of the larger triangle.

\[
\frac{AB}{AD} = \frac{BG}{DE}
\]

Here,
- \( AB = 16 \, \text{m} \)
- \( AD = 16 + 20 + 24 = 60 \, \text{m} \)
- \( DE = 45 \, \text{m} \)

Thus:

\[
\frac{16}{60} = \frac{X}{45}
\]

Solving for \( X \):

\[
X = \frac{16 \times 45}{60} = 12 \, \text{m}
\]

#### 2. Set up a proportion for \( Y = CF \):

We now use similar triangles \( \triangle ACD \) and \( \triangle AED \). Again, we apply the proportion of sides:

\[
\frac{AC}{AD} = \frac{CF}{DE}
\]

Here,
- \( AC = 16 + 20 = 36 \, \text{m} \)
- \( AD = 60 \, \text{m} \)
- \( DE = 45 \, \text{m} \)

Thus:

\[
\frac{36}{60} = \frac{Y}{45}
\]

Solving for \( Y \):

\[
Y = \frac{36 \times 45}{60} = 27 \, \text{m}
\]

### Final Answers:
- \( X = BG = 12 \, \text{m} \)
- \( Y = CF = 27 \, \text{m} \)

Would you like further clarification or a more detailed explanation of any step? 

Here are some related questions:
1. What are the properties of similar triangles used in this problem?
2. How do you calculate the area of a triangle given its base and height?
3. How could you verify the similarity of the triangles in this diagram?
4. What would happen if one of the triangle’s proportions was altered?
5. How can we apply the Pythagorean theorem to find diagonal lengths in this context?

**Tip:** In problems with similar triangles, setting up correct proportions is key to finding unknown sides. Always make sure to match corresponding sides properly.

Determine the lengths of BG and CF in these similar triangles.

Learn how to determine the lengths of segments BG and CF in similar triangles by applying the properties of proportions. This problem involves using triangle similarity theorem and is suitable for Grades 9-11 students.

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