We are given a triangle $\triangle ABC$ with:

• $BD = 30 \, m$
• $\angle DBC = 90^\circ$
• $\angle ACB = 40^\circ$

We are tasked with calculating the following sides:

• $AB$
• $BC$
• $AD$
• $DC$
• $AC$

Step-by-Step Solution:

1. Interpretation:

   • $BD$ is perpendicular to $AC$, so $\triangle BDC$ is a right triangle.
   • $\angle ACB = 40^\circ$.

2. Using Trigonometry: Since $\triangle ABC$ is a triangle with one known angle $\angle ACB = 40^\circ$, we can calculate the unknown sides using trigonometric ratios. First, we need to calculate the length of $BC$ (adjacent side to $\angle ACB$).

3. Calculate $BC$: Using trigonometric identities: $\tan(40^\circ) = \frac{BD}{BC}$ Solving for $BC$: $BC = \frac{BD}{\tan(40^\circ)} = \frac{30}{\tan(40^\circ)}$

4. Use Pythagoras Theorem in $\triangle BDC$: After calculating $BC$, we can use Pythagoras theorem in $\triangle BDC$ to find $DC$.

Let me calculate these values now.We have calculated the following:

• $BC = 35.75 \, \text{m}$
• $DC = 19.45 \, \text{m}$

Next, let's calculate the remaining sides $AB$, $AD$, and $AC$.

1. Calculate $AB$: Using the sine rule or cosine rule based on the angles, we will calculate $AB$.

2. Calculate $AC$: Using the trigonometric ratios, we can determine $AC$, which will be the hypotenuse in $\triangle ABC$.

Let me compute these now.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, based on the previous calculations:

• $BC = 35.75 \, \text{m}$
• $DC = 19.45 \, \text{m}$

Would you like help finishing the rest of the problem manually, or do you have any questions so far?

Here are some related questions to think about:

1. How can we use the sine rule to calculate unknown sides in triangles?
2. What is the relationship between the angles and sides in a right triangle?
3. How do we apply the Pythagorean theorem in different right triangles?
4. Can you find $AB$ and $AC$ using trigonometric functions like sine and cosine?
5. What other methods can be used to solve a triangle with known angles and sides?

Tip: Always remember to use appropriate trigonometric identities depending on the known angles and sides to solve for unknown parts of the triangle.