https://file.aiquickdraw.com/mathbot/file/17254788687937ufgdzgo.jpg

The problem in the image reads as follows:

*"In a parallelogram ABCD, the side \( AB = 20 \) meters determines an angle \( \theta = 60^\circ \) with the side \( AD \), which has a length of 10 meters. Find: a) the area of the triangle \( ABD \), b) the area of the parallelogram."*

Let's solve this step by step.

### Part a) Area of Triangle \( ABD \)

To find the area of triangle \( ABD \), we can use the formula for the area of a triangle given two sides and the included angle:

\[
\text{Area} = \frac{1}{2} \times AB \times AD \times \sin(\theta)
\]

Here:
- \( AB = 20 \) meters
- \( AD = 10 \) meters
- \( \theta = 60^\circ \)

Substituting the values:

\[
\text{Area} = \frac{1}{2} \times 20 \times 10 \times \sin(60^\circ)
\]

The value of \( \sin(60^\circ) \) is \( \frac{\sqrt{3}}{2} \). Therefore:

\[
\text{Area} = \frac{1}{2} \times 20 \times 10 \times \frac{\sqrt{3}}{2} = 100 \times \frac{\sqrt{3}}{2} = 50\sqrt{3} \text{ square meters}
\]

### Part b) Area of the Parallelogram

The area of the parallelogram can be found using the formula:

\[
\text{Area of parallelogram} = AB \times AD \times \sin(\theta)
\]

Substituting the values:

\[
\text{Area of parallelogram} = 20 \times 10 \times \sin(60^\circ) = 200 \times \frac{\sqrt{3}}{2} = 100\sqrt{3} \text{ square meters}
\]

### Final Answer:
- **a)** The area of triangle \( ABD \) is \( 50\sqrt{3} \) square meters.
- **b)** The area of the parallelogram is \( 100\sqrt{3} \) square meters.

Would you like more details or have any questions about these calculations?

Here are five related questions:
1. What is the significance of the sine function in calculating the area of a triangle?
2. How would the area change if the angle \( \theta \) were 90 degrees?
3. What other methods can be used to calculate the area of a parallelogram?
4. How does the area of the triangle relate to the area of the parallelogram?
5. Can the area of the triangle be calculated using vectors? If so, how?

**Tip:** Always remember that the area of a parallelogram is double that of a triangle formed by one of its diagonals.

In a parallelogram ABCD, the side AB = 20 meters determines an angle θ = 60° with the side AD, which has a length of 10 meters. Find: a) the area of the triangle ABD, b) the area of the parallelogram.

Learn how to calculate the area of triangle ABD and parallelogram ABCD given the sides and the angle between them. This problem involves key trigonometric concepts and formulas, suitable for students in Grades 9-12.

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