If the sides of the parallelogram and an includes angle are 9 m, 13 m, and 150⁰, respectively. Find
the length of the shorter diagonal and the area of the parallelogram.

To find the length of the shorter diagonal and the area of the parallelogram, we'll use the following formulas:

1. **Length of the Diagonal**:
   For a parallelogram with sides \( a \) and \( b \) and an included angle \( \theta \), the length of one diagonal \( d_1 \) can be calculated using the cosine rule (law of cosines):

   \[
   d_1 = \sqrt{a^2 + b^2 - 2ab\cos(\theta)}
   \]

2. **Area of the Parallelogram**:
   The area \( A \) of the parallelogram can be calculated using the formula:

   \[
   A = ab \sin(\theta)
   \]

Given:
- \( a = 9 \, \text{m} \)
- \( b = 13 \, \text{m} \)
- \( \theta = 150^\circ \)

### Step 1: Calculating the shorter diagonal \( d_1 \)
We will use the cosine rule to calculate \( d_1 \). First, convert \( \theta \) from degrees to radians, if needed, and compute the cosine of \( 150^\circ \).

### Step 2: Calculating the area of the parallelogram \( A \)
We'll use the formula for the area directly by computing the sine of \( 150^\circ \).

Let's calculate these values.The length of the shorter diagonal of the parallelogram is approximately \( 21.28 \, \text{m} \), and the area of the parallelogram is approximately \( 58.5 \, \text{m}^2 \).

Would you like further details or explanations? 

Here are some related questions to consider:
1. How would the area change if the included angle were \( 90^\circ \)?
2. What is the length of the longer diagonal?
3. How does the law of cosines apply to parallelograms generally?
4. Can you determine the height of the parallelogram using this information?
5. How do the sine and cosine rules help in non-right-angled triangles and parallelograms?

**Tip**: When working with angles in trigonometry, always check if your calculator is in degrees or radians mode, depending on the angle measure!

If the sides of the parallelogram and an included angle are 9 m, 13 m, and 150⁰, respectively. Find the length of the shorter diagonal and the area of the parallelogram.

Learn how to find the shorter diagonal and area of a parallelogram with sides 9 m and 13 m, and an included angle of 150⁰. Using the law of cosines and sine rule, this problem is suitable for students in Grades 10-12. Key formulas include diagonal length calculation and the area of a parallelogram.

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