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We are given a triangle \( ABC \) with points \( D \) on side \( AC \) and \( E \) on side \( AB \), and a straight line \( DE \). Our task is to find the length of side \( CD \).

### Information given:
- \( AD = 27 \, \text{m} \)
- \( AE = 39 \, \text{m} \)
- \( DE = 23 \, \text{m} \)
- \( BC = 65 \, \text{m} \)
- \( \angle ABC = 68^\circ \)

The solution involves using the **sine rule** in triangles and potentially the **cosine rule**.

### Approach:
1. First, we observe two triangles: \( \triangle ABC \) and \( \triangle ADE \). The information given focuses more on \( \triangle ABC \).
2. Use the **sine rule** to relate the sides and angles in \( \triangle ABC \). We know:
   \[
   \frac{AB}{\sin(\angle ACB)} = \frac{BC}{\sin(\angle BAC)} = \frac{AC}{\sin(\angle ABC)}
   \]

   We are missing some key angles but we know that \( \angle ABC = 68^\circ \). To use the sine rule, we might need to calculate or estimate some angles or side lengths.

3. We may also use the **cosine rule** in \( \triangle ABC \) since we have two sides and the included angle:
   \[
   AC^2 = AB^2 + BC^2 - 2 \times AB \times BC \times \cos(\angle ABC)
   \]

Let's work this out:

- \( AB = AD + DE + AE = 27 + 23 + 39 = 89 \, \text{m} \)
- Now apply the cosine rule to find \( AC \):
  \[
  AC^2 = 89^2 + 65^2 - 2 \times 89 \times 65 \times \cos(68^\circ)
  \]
  \[
  AC^2 = 7921 + 4225 - 2 \times 89 \times 65 \times 0.3746
  \]
  \[
  AC^2 = 12146 - 4335.67 = 7810.33
  \]
  \[
  AC = \sqrt{7810.33} \approx 88.38 \, \text{m}
  \]

4. Now that we have \( AC \), we can find \( CD \) by subtracting \( AD \) from \( AC \):
   \[
   CD = AC - AD = 88.38 - 27 = 61.38 \, \text{m}
   \]

Thus, the length of \( CD \) is approximately \( 61.38 \, \text{m} \).

Would you like more details or have any questions? 

Here are 5 related questions:
1. How do we apply the sine rule in non-right-angled triangles?
2. What is the significance of the cosine rule in triangle geometry?
3. Can you explain how angle-side relationships affect triangle calculations?
4. What would be a step-by-step guide for using the cosine rule?
5. How can we apply trigonometry to solve problems involving multiple triangles?

**Tip**: When solving for side lengths in non-right-angled triangles, always check if the sine or cosine rule applies based on the available data.

The diagram shows triangle ABC with D on AC and E on AB. DE is a straight line. AD = 27 m, AE = 39 m, DE = 23 m, and BC = 65 m. Angle ABC = 68°. Calculate the length of CD.

Solve for the length of CD in triangle ABC using the sine and cosine rules. This problem involves calculating side lengths based on given angles and side measurements, with data such as AD = 27 m, AE = 39 m, DE = 23 m, and BC = 65 m. A comprehensive step-by-step guide suitable for students in Grades 9-11.

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