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### (a) **Perimeter of the field ABCDE:**

The perimeter is the sum of the lengths of all the sides of the field. Based on the diagram:
- \(AB = 100 \, \text{m}\)
- \(BC = 170 \, \text{m}\)
- \(CD = 120 \, \text{m}\)
- \(DE = 50 \, \text{m}\)
- \(EA = 130 \, \text{m}\)

Now we sum the sides:
\[
\text{Perimeter} = AB + BC + CD + DE + EA
\]
\[
\text{Perimeter} = 100 + 170 + 120 + 50 + 130 = 570 \, \text{m}
\]

Thus, the perimeter of the field is **570 m**.

### (b) **Calculate angle ABD:**

We can use the **cosine rule** to find angle \(ABD\), since we know the lengths of sides \(AB = 100 \, \text{m}\), \(BD = 150 \, \text{m}\), and \(AD = 120 \, \text{m}\).

The cosine rule states:
\[
\cos(\theta) = \frac{a^2 + b^2 - c^2}{2ab}
\]
Where:
- \(a = AB = 100 \, \text{m}\)
- \(b = BD = 150 \, \text{m}\)
- \(c = AD = 120 \, \text{m}\)

Substituting the values:
\[
\cos(\theta) = \frac{100^2 + 150^2 - 120^2}{2 \times 100 \times 150}
\]
\[
\cos(\theta) = \frac{10000 + 22500 - 14400}{30000}
\]
\[
\cos(\theta) = \frac{18100}{30000} \approx 0.6033
\]

Now, using the inverse cosine function:
\[
\theta = \cos^{-1}(0.6033) \approx 52.9^\circ
\]

Thus, angle \(ABD\) is approximately **52.9°**.

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Would you like further clarification or any additional details on this solution? Here are five related questions that might interest you:

1. How would you find the area of the field ABCDE using the known lengths?
2. What if one of the angles in the triangle was unknown? How would we approach that?
3. Can you verify angle ABD using the sine rule?
4. How could the Pythagorean theorem apply to certain triangles in the field?
5. What role does the cosine rule play in more complex geometrical problems?

**Tip:** When solving for angles using trigonometric rules, double-check your values with inverse functions to ensure precision!

Calculate the perimeter of the field ABCDE and angle ABD using the given lengths.

Learn how to calculate the perimeter of a field ABCDE and angle ABD using trigonometry and geometry principles. This problem involves the use of the cosine rule and is suitable for students in Grades 9-10.

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