The diagonal of a rectangle is 8 cm. The angle between side and diagonal of a rectangle is 30°, then the area of rectangle is :

Question

Accepted Answer

To determine the area of the rectangle given the diagonal and the angle between the side and the diagonal, we can use trigonometric relationships.

1. **Understanding the Problem:**
   - The diagonal of the rectangle is given as 8 cm.
   - The angle between the side and the diagonal is 30°.

2. **Using Trigonometry:**
   - In a right triangle formed by the diagonal and the sides of the rectangle, the diagonal acts as the hypotenuse.
   - Let's denote the sides of the rectangle as $ a $ and $ b $.
   - The angle between side $ a $ and the diagonal is 30°.

3. **Calculating the Sides:**
   - Using the cosine function for the angle:
     $$
     \cos(30°) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{a}{8}
     $$
     $$
     a = 8 \cdot \cos(30°) = 8 \cdot \frac{\sqrt{3}}{2} = 4\sqrt{3} \text{ cm}
     $$

- Using the sine function for the angle:
     $$
     \sin(30°) = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{b}{8}
     $$
     $$
     b = 8 \cdot \sin(30°) = 8 \cdot \frac{1}{2} = 4 \text{ cm}
     $$

4. **Calculating the Area:**
   - The area of the rectangle is given by the product of its sides:
     $$
     \text{Area} = a \cdot b = (4\sqrt{3}) \cdot 4 = 16\sqrt{3} \text{ cm}^2
     $$

5. **Justification for Option 2:**
   - The calculated area of the rectangle is $ 16\sqrt{3} \text{ cm}^2 $, which matches Option 2.

6. **Explanation for Incorrect Options:**
   - **Option 1: $16 \text{ cm}^2$** - This option does not account for the trigonometric calculations involving the angle and the diagonal.
   - **Option 3: $8\sqrt{3} \text{ cm}^2$** - This option underestimates the area by not correctly applying the product of the sides.
   - **Option 4: $\frac{16}{\sqrt{3}} \text{ cm}^2$** - This option incorrectly applies the trigonometric relationships and does not result in the correct area.

Therefore, based on the given information and calculations, Option 2 is indeed the correct answer.