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 solve this problem, we need to find two things:
1. The length of the shorter diagonal of the parallelogram.
2. The area of the parallelogram.

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

### Step 1: Finding the Length of the Shorter Diagonal

We can use the **diagonal formula** for a parallelogram to find the length of the diagonals. The formula for the diagonals is based on the law of cosines:

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

where \( d_1 \) is one of the diagonals and \( \theta \) is the included angle. Since we want the shorter diagonal, we’ll compute this first.

Substituting the values into the equation:
\[
d_1 = \sqrt{9^2 + 13^2 - 2 \cdot 9 \cdot 13 \cdot \cos(150^\circ)}
\]

First, calculate the cosine of \( 150^\circ \):
\[
\cos(150^\circ) = -\cos(30^\circ) = -\frac{\sqrt{3}}{2} \approx -0.866
\]

Now, substitute this into the formula:
\[
d_1 = \sqrt{9^2 + 13^2 - 2 \cdot 9 \cdot 13 \cdot (-0.866)}
\]
\[
d_1 = \sqrt{81 + 169 + 2 \cdot 9 \cdot 13 \cdot 0.866}
\]
\[
d_1 = \sqrt{250 + 202.716}
\]
\[
d_1 = \sqrt{452.716} \approx 21.27 \, \text{m}
\]

Thus, the length of the shorter diagonal is approximately \( 21.27 \, \text{m} \).

### Step 2: Finding the Area of the Parallelogram

The formula for the area of a parallelogram is given by:

\[
\text{Area} = a \cdot b \cdot \sin(\theta)
\]

Substituting the values into the equation:
\[
\text{Area} = 9 \cdot 13 \cdot \sin(150^\circ)
\]

Since \( \sin(150^\circ) = \sin(30^\circ) = 0.5 \), we get:
\[
\text{Area} = 9 \cdot 13 \cdot 0.5 = 58.5 \, \text{m}^2
\]

### Final Answers:
- The length of the shorter diagonal is approximately \( 21.27 \, \text{m} \).
- The area of the parallelogram is \( 58.5 \, \text{m}^2 \).

Would you like more details on any step? Here are five related questions to extend this topic:

1. How would you find the longer diagonal of the parallelogram?
2. What if the included angle was \( 120^\circ \)? How would the calculations change?
3. Can you calculate the perimeter of the parallelogram?
4. How does the shape of the parallelogram change as the included angle increases?
5. How would you compute the height of the parallelogram?

**Tip:** To compute diagonals in parallelograms, the law of cosines is especially useful because it works well when you know two sides and an included angle.

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 length of the shorter diagonal and the area of a parallelogram with sides 9m and 13m, and an included angle of 150°. Step-by-step solution using the Law of Cosines and sine functions. Suitable for high school geometry students.

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