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.