https://file.aiquickdraw.com/mathbot/file/1724887320170zza3j7u4.jpg

To solve the problem in the image, we need to analyze the information provided and apply relevant mathematical concepts. The problem involves a triangle \( ABC \) with points \( D \) on \( AC \) and \( E \) on \( AB \). A line \( DE \) is drawn.

### Given:
- \( AD = 28 \) m
- \( AE = 41 \) m
- \( DE = 28 \) m
- \( BC = 84 \) m
- \( \angle EDB = 72^\circ \)
- \( \angle DBC = 64^\circ \)

#### (a) **Calculate the size of angle \( DAE \):**

We can use the fact that the angles in a triangle sum to \( 180^\circ \).

1. First, let's calculate angle \( \angle EBD \) in triangle \( EBD \).
    \[
    \angle EBD = 180^\circ - 72^\circ = 108^\circ
    \]
   
2. Since \( \angle DBC = 64^\circ \), we can now find \( \angle DBE \) by subtracting from \( \angle EBD \):
    \[
    \angle DBE = 108^\circ - 64^\circ = 44^\circ
    \]

3. To find angle \( \angle DAE \), recognize that \( \angle DAE \) and \( \angle EDB \) are related by the fact that \( DAE \) is an external angle to triangle \( EBD \):
    \[
    \angle DAE = 180^\circ - \angle DBE = 180^\circ - 44^\circ = 136^\circ
    \]

However, this approach needs verification as it involves angles not directly tied by simple triangle relationships.

#### (b) **Calculate the length of \( CD \):**

1. Since \( BC = 84 \) m and triangle \( ABC \) is not necessarily right-angled or equilateral, we may need additional methods such as the cosine rule to find \( CD \).

The cosine rule for triangle \( BCD \) gives:
\[
CD^2 = BC^2 + BD^2 - 2 \times BC \times BD \times \cos(\angle DBC)
\]
Substituting the known values into the formula:
\[
CD^2 = 84^2 + BD^2 - 2 \times 84 \times BD \times \cos(64^\circ)
\]
The length of \( BD \) needs to be determined or estimated from previous steps to solve this equation.

### Final Steps:
1. Calculate angle \( DAE \) correctly by double-checking relationships.
2. Confirm length \( CD \) with accurate trigonometric methods.

Would you like a detailed step-by-step solution or assistance with any specific part of the problem? Here are 5 related questions:

1. How does the sine rule apply to non-right-angled triangles?
2. What is the relationship between internal and external angles in triangles?
3. How is the cosine rule derived and when is it most useful?
4. Why is it important to verify calculations involving angles in geometric problems?
5. What are the common mistakes to avoid when dealing with non-right triangles?

**Tip:** Always draw a clear diagram and label all known values to help visualize the problem effectively.

Learn how to solve a complex triangle geometry problem involving angles and lengths. This problem includes using the cosine rule to find lengths and angles within a triangle. Suitable for high school students studying geometry and trigonometry.

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