We have a paralelogram.
We know diagonal e = 20 cm.
Side a = 12cm.
The diagonals form angles of 30 and 150. 
30° is above side a.

Find side b and calculate the area of this paralelogram.

To solve this problem, let’s go step-by-step. Here’s what we know about the parallelogram:

1. **Diagonal \( e \) = 20 cm**
2. **Side \( a \) = 12 cm**
3. **The diagonals form angles of 30° and 150°**, with 30° being the angle between diagonal \( e \) and side \( a \).

To find side \( b \) and the area of the parallelogram, we can proceed as follows:

### Step 1: Find side \( b \) using the law of cosines

In any parallelogram, the diagonals bisect each other, forming two triangles within the parallelogram. In this problem, the given diagonal \( e \) forms an angle of 30° with side \( a \). We can use the **law of cosines** in the triangle that includes \( a \), \( b \), and the half-lengths of the diagonals.

The formula for the law of cosines is:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(\theta)
\]
where:
- \( c \) is the length of the diagonal (20 cm),
- \( a \) is one side of the parallelogram (12 cm),
- \( b \) is the other side of the parallelogram (which we need to find), and
- \( \theta = 30^\circ \) (angle between \( a \) and \( e \)).

Rearrange the formula to solve for \( b \):
\[
b = \sqrt{\frac{e^2 - a^2 + 2a \cdot e \cdot \cos(30^\circ)}{2}}
\]

### Step 2: Calculate

We have a parallelogram. We know diagonal e = 20 cm. Side a = 12 cm. The diagonals form angles of 30° and 150°. 30° is above side a. Find side b and calculate the area of this parallelogram.

Learn how to find side b and calculate the area of a parallelogram with a diagonal of 20 cm, side of 12 cm, and angles of 30° and 150°. This problem uses the law of cosines and area calculation formulas, ideal for students in Grades 10-12.

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