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### Problem Breakdown:

The diagram represents a quadrilateral \(ABCD\) with the following measurements provided:
- \(AB = 14\)
- \(AC = 12\)
- \(CD = 7\)

The goal is to:
1. **Calculate the length of \(AD\)**.
2. **Determine the value of \(\theta\)**.

### Part i: Calculate the Length of \(AD\)

Looking at the diagram, \(AC\) is a diagonal and \(AD\) forms part of a triangle. Since this involves a triangle, we likely need to use the **Cosine Rule** because the angles involved aren't necessarily right angles, except where noted (at point \(C\)).

The Cosine Rule is given by:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(\theta)
\]

Here, \(c = AD\), \(a = AB = 14\), \(b = AC = 12\), and \(\theta\) is the angle at \(C\), which can be found or given later.

#### Step 1: Applying the Pythagorean Theorem (if applicable)
If we consider \( \triangle ACD \), and if \(\angle ACD = 90^\circ\) (which seems to be the case according to the right angle marked in the diagram), then we can use the Pythagorean Theorem.

\[
AD^2 = AC^2 + CD^2
\]
\[
AD^2 = 12^2 + 7^2
\]
\[
AD^2 = 144 + 49 = 193
\]
\[
AD = \sqrt{193} \approx 13.89
\]

So, \(AD \approx 13.89\) units.

### Part ii: Determine the Value of \(\theta\)

Given that \(\theta\) is likely at \(\angle ABD\) (from the diagram), and using trigonometric relationships:

Using the sine rule or other trigonometric identities might be necessary depending on further context or specific requirements (like angle measures).

However, further details or values would be needed for a precise determination of \(\theta\).

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Would you like more details on the determination of \(\theta\) or any other aspect of this problem?

### 5 Relative Questions:
1. How do we apply the Sine Rule in non-right angled triangles?
2. What are the conditions to use the Cosine Rule?
3. How can we determine angles if only side lengths are provided?
4. What is the importance of the Pythagorean Theorem in solving such problems?
5. Can we use vector analysis for solving these types of geometric problems?

### Tip:
When dealing with triangles, always identify if it's a right-angled triangle first, as it allows the use of simpler trigonometric relationships like the Pythagorean Theorem.

In the diagram below, AB = 14, AC = 12, and CD = 7. Calculate the length of AD and determine the value of θ.

Learn how to calculate the length of AD in a geometric figure with given sides and angles using the Pythagorean theorem and Cosine Rule. This problem explores key geometry and trigonometry concepts suitable for students in Grades 10-12.

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