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The problem asks for the perimeter of a quadrilateral-shaped building site. The diagram provides the following details:

- Side lengths: \( 28.5 \, \text{m} \), \( 52.4 \, \text{m} \), and \( 84.6 \, \text{m} \)
- Interior angles: \( 72^\circ \) and \( 75^\circ \)

To find the perimeter of the quadrilateral, we need to determine the length of the fourth side.

### Steps to solve:

1. **Use the Cosine Rule** to find the length of the fourth side.
   
   The Cosine Rule for a triangle states:
   \[
   c^2 = a^2 + b^2 - 2ab \cos(C)
   \]
   where:
   - \( c \) is the side opposite angle \( C \),
   - \( a \) and \( b \) are the other two sides,
   - \( C \) is the angle between sides \( a \) and \( b \).

2. **Apply the Cosine Rule** to one of the triangles within the quadrilateral:
   - Consider the triangle formed by the sides \( 28.5 \, \text{m} \), \( 84.6 \, \text{m} \), and the angle \( 72^\circ \).
   - Let the fourth side of the quadrilateral be \( x \).

   \[
   x^2 = 28.5^2 + 84.6^2 - 2(28.5)(84.6)\cos(72^\circ)
   \]
   Calculate \( x \) from this.

3. **Find the perimeter** by summing all four sides.

I'll calculate the exact value now.The length of the fourth side is approximately \( 80.49 \, \text{m} \).

### Now, to find the perimeter:
- \( 28.5 \, \text{m} \)
- \( 52.4 \, \text{m} \)
- \( 84.6 \, \text{m} \)
- \( 80.49 \, \text{m} \)

The perimeter \( P \) is the sum of all these sides.

\[
P = 28.5 + 52.4 + 84.6 + 80.49 = 245.99 \, \text{m}
\]

Thus, the perimeter of the quadrilateral is approximately \( 245.99 \, \text{m} \).

Would you like more details or have any questions?

### Related Questions:
1. How would the perimeter change if one of the interior angles was different?
2. What is the area of this quadrilateral?
3. How can you determine if this quadrilateral is cyclic?
4. What are some real-world applications of using the cosine rule in construction?
5. How would the perimeter be calculated if the site was a regular polygon?

### Tip:
When solving geometry problems involving quadrilaterals, breaking them into triangles can simplify the use of trigonometric rules like the cosine rule.

A building site is in the form of a quadrilateral as shown below. Determine the length of the perimeter of the site.

Learn how to determine the perimeter of a quadrilateral-shaped building site using the Cosine Rule. This problem involves applying trigonometric concepts and formulas, including the cosine rule for calculating unknown side lengths in a triangle. Suitable for students in Grades 9-12.

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