In a parallelogram, the ratio of the two adjacent sides is 1:2. If the area of the parallelogram is 36√2 square unit and the angle between the two sides is 45°, what is the area, in square unit, of the rectangle whose smaller side is equal to the smaller side of the parallelogram and the larger side is equal to the larger side of the parallelogram?

Given:
- The ratio of the two adjacent sides of a parallelogram is 1:2.
- The area of the parallelogram is \( 36\sqrt{2} \) square units.
- The angle between the two sides is \( 45^\circ \).

Let's denote the sides of the parallelogram as \( a \) and \( 2a \), where \( a \) is the length of the smaller side and \( 2a \) is the length of the larger side.

### Step-by-Step Solution

1. **Area of Parallelogram:**
   The area \( A \) of a parallelogram is given by the formula:
   \[
   A = ab \sin \theta
   \]
   where \( a \) and \( b \) are the lengths of the adjacent sides, and \( \theta \) is the angle between them.

2. **Substitute Given Values:**
   Here, \( a \) is the smaller side, \( 2a \) is the larger side, and \( \theta = 45^\circ \). So,
   \[
   36\sqrt{2} = a \cdot 2a \cdot \sin 45^\circ
   \]

3. **Simplify the Equation:**
   \[
   36\sqrt{2} = 2a^2 \cdot \sin 45^\circ
   \]
   We know that \( \sin 45^\circ = \frac{\sqrt{2}}{2} \). Thus,
   \[
   36\sqrt{2} = 2a^2 \cdot \frac{\sqrt{2}}{2}
   \]
   Simplifying further:
   \[
   36\sqrt{2} = a^2 \sqrt{2}
   \]

4. **Solve for \( a \):**
   \[
   36\sqrt{2} = a^2 \sqrt{2}
   \]
   Divide both sides by \( \sqrt{2} \):
   \[
   36 = a^2
   \]
   Thus,
   \[
   a = \sqrt{36} = 6
   \]

5. **Determine the Sides of the Rectangle:**
   The smaller side of the parallelogram is \( 6 \) units (which is \( a \)), and the larger side is \( 2a = 12 \) units.

6. **Area of the Rectangle:**
   The area \( A_{\text{rect}} \) of the rectangle whose sides are \( 6 \) units and \( 12 \) units is:
   \[
   A_{\text{rect}} = 6 \times 12 = 72 \text{ square units}
   \]

Therefore, the area of the rectangle is \( 72 \) square units.

### Do you want details or have any questions?

### Related Questions:

1. How do you derive the formula for the area of a parallelogram?
2. What other properties of parallelograms are useful in geometry?
3. Can you explain the significance of the angle in calculating the area of a parallelogram?
4. How does the sine function affect the area calculation of a parallelogram?
5. What is the relationship between parallelograms and rectangles in terms of area?
6. How do you find the area of a parallelogram if the angle is not given?
7. Can you show a geometric proof of why the area formula \( A = ab \sin \theta \) works for parallelograms?
8. How would the area change if the angle between the sides was different?

### Tip:

Always remember to verify the trigonometric values of angles, especially for common angles like \( 45^\circ \), \( 30^\circ \), and \( 60^\circ \), as they are frequently used in geometry problems.

Learn how to calculate the area of a parallelogram with sides in the ratio 1:2, an area of 36√2 square units, and an included angle of 45°. Detailed step-by-step solution included. Suitable for high school geometry students.

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