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We are given a diagram with two triangles, \( \triangle ABC \) and \( \triangle ACD \), where \( BC = 7 \, \text{m} \) and various angles are provided. The angles given are:

- \( \angle ABC = 35^\circ \)
- \( \angle ACB = 67^\circ \)
- \( \angle ACD = 60^\circ \)

To solve for the lengths of \( AB \), \( AC \), and \( AD \), we will use the **Law of Sines**, which states:

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

### Step 1: Solving for \( AB \) and \( AC \) in \( \triangle ABC \)
The angles in a triangle sum up to \( 180^\circ \), so:

\[
\angle BAC = 180^\circ - 35^\circ - 67^\circ = 78^\circ
\]

Now we apply the Law of Sines to \( \triangle ABC \):

\[
\frac{AB}{\sin 67^\circ} = \frac{AC}{\sin 35^\circ} = \frac{BC}{\sin 78^\circ}
\]

We know \( BC = 7 \, \text{m} \), so:

\[
\frac{7}{\sin 78^\circ} = \frac{AB}{\sin 67^\circ}
\]

Solving for \( AB \):

\[
AB = \frac{7 \cdot \sin 67^\circ}{\sin 78^\circ}
\]
\[
AB \approx \frac{7 \cdot 0.9205}{0.9781} \approx 6.59 \, \text{m}
\]

Similarly, for \( AC \):

\[
\frac{7}{\sin 78^\circ} = \frac{AC}{\sin 35^\circ}
\]
\[
AC = \frac{7 \cdot \sin 35^\circ}{\sin 78^\circ}
\]
\[
AC \approx \frac{7 \cdot 0.5736}{0.9781} \approx 4.10 \, \text{m}
\]

### Step 2: Solving for \( AD \) in \( \triangle ACD \)
Now consider \( \triangle ACD \). Since we know \( AC \), we can apply the Law of Sines again. First, we need to find \( \angle CAD \). Since \( \angle ACD = 60^\circ \) and \( \angle ACB = 67^\circ \):

\[
\angle CAD = 180^\circ - 60^\circ - 67^\circ = 53^\circ
\]

Now apply the Law of Sines in \( \triangle ACD \):

\[
\frac{AD}{\sin 67^\circ} = \frac{AC}{\sin 60^\circ}
\]

Substitute the known value of \( AC \approx 4.10 \, \text{m} \):

\[
AD = \frac{4.10 \cdot \sin 67^\circ}{\sin 60^\circ}
\]
\[
AD \approx \frac{4.10 \cdot 0.9205}{0.8660} \approx 4.36 \, \text{m}
\]

### Final Results:
- \( AB \approx 6.59 \, \text{m} \)
- \( AC \approx 4.10 \, \text{m} \)
- \( AD \approx 4.36 \, \text{m} \)

Would you like further details or clarification on any of the steps? 

Here are 5 follow-up questions:
1. How would the Law of Cosines be applied to this problem instead of the Law of Sines?
2. What is the significance of knowing all the angles in both triangles?
3. Can you calculate the area of \( \triangle ABC \) using the calculated sides?
4. How could you verify the result of \( AD \) through another method?
5. How would the solution change if \( BC \) were a different length?

**Tip**: When solving for unknown sides in non-right triangles, the Law of Sines is often the simplest method, especially when two angles and one side are known.

find AB, AC, and AD in the given triangle configuration

Learn how to solve for the sides AB, AC, and AD in a triangle using the Law of Sines. The problem involves applying the Law of Sines and trigonometric concepts to find unknown side lengths based on given angles and side lengths. Suitable for Grades 10-12.

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