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The image you provided presents a trigonometry problem involving a non-right angled triangle. Let's break it down step by step, as it outlines three tasks that involve using the **cosine rule**, verifying using **sine rule**, and then applying **Heron's formula** to calculate the area.

### Given:
- The triangle has sides of lengths 50 meters, 60 meters, and 70 meters.

### Task 1: Use the cosine rule to calculate the largest angle of the triangle.

The **Cosine Rule** is given as:

\[
\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}
\]

Where \( C \) is the angle opposite the largest side (here, the side of 70 meters), and \( a = 50 \) meters, \( b = 60 \) meters, and \( c = 70 \) meters.

We will calculate the largest angle using this formula.

### Task 2: Verify your answer using the sine rule to calculate one of the remaining angles.

The **Sine Rule** states:

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

After finding the largest angle \( C \), we can use this formula to verify the other angles.

### Task 3: Calculate the area of the triangle using both Heron’s formula and \( \frac{1}{2}ab\sin(C) \).

- **Heron's Formula** for the area is given by:

\[
A = \sqrt{s(s - a)(s - b)(s - c)}
\]
Where \( s = \frac{a + b + c}{2} \) is the semi-perimeter of the triangle.

- The second method to calculate the area uses:

\[
A = \frac{1}{2}ab\sin(C)
\]

Now, I will calculate the largest angle, verify it, and find the area using both methods. Let's compute these step by step.### Task 1: Largest Angle Using the Cosine Rule
- The largest angle \( C \) opposite the 70-meter side is approximately **78.46°**.

### Task 2: Verifying Using the Sine Rule
- The other angles are:
  - \( A \) opposite the 50-meter side is approximately **44.42°**.
  - \( B \) opposite the 60-meter side is approximately **57.12°**.

### Task 3: Area Calculation
- Using **Heron’s Formula**, the area of the triangle is approximately **1469.69 square meters**.
- Using \( \frac{1}{2}ab\sin(C) \), the area is also **1469.69 square meters**, confirming the result.

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. What if the triangle had different side lengths? How would the process change?
2. How do you derive the cosine rule and sine rule from basic geometry principles?
3. Can you apply the sine rule to find all angles if no angle is given initially?
4. How does Heron’s formula work if the triangle is nearly equilateral?
5. What other methods can you use to find the area of non-right-angled triangles?

**Tip:** Always double-check that the sum of angles in any triangle adds up to exactly 180°.

A surveyor is mapping a triangular plot of land. The sides of the triangle measure 50 meters, 60 meters, and 70 meters. Task 1: Use the cosine rule to calculate the largest angle of the triangle. Task 2: Verify your answer by using the sine rule to calculate one of the remaining angles. Task 3: Calculate the area of the triangle using both Heron’s formula and the area formula with sine.

Solve a trigonometry problem involving a triangular plot of land with sides 50 meters, 60 meters, and 70 meters. Learn how to apply the cosine rule, sine rule, and Heron's formula to calculate angles and the area of the triangle. Detailed step-by-step solution provided, ideal for Grades 10-12.

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