A square and a rhombus have the same base and rhombus is inclined at `30^(@)`. What is the ratio of area of the square to the area of the rhombus.

Options

A. `sqrt(2):1`
B. `2:1`
C. `1:1`
D. `2:sqrt(3)`

Answer

To find the ratio of the area of a square to the area of a rhombus that has the same base and is inclined at \(30^\circ\), we can follow these steps:

### Step 1: Define the side length
Let the side length of both the square and the rhombus be \(A\).

### Step 2: Calculate the area of the square
The area of a square is given by the formula:
\[ \text{Area of Square} = A^2 \]

### Step 3: Calculate the area of the rhombus
The area of a rhombus can be calculated using the formula:
\[ \text{Area of Rhombus} = \text{Base} \times \text{Height} \]
Since the rhombus is inclined at \(30^\circ\), we can find the height using the sine function:
\[ \text{Height} = A \cdot \sin(30^\circ) = A \cdot \frac{1}{2} = \frac{A}{2} \]
Thus, the area of the rhombus becomes:
\[ \text{Area of Rhombus} = A \cdot \frac{A}{2} = \frac{A^2}{2} \]

### Step 4: Find the ratio of the areas
Now, we can find the ratio of the area of the square to the area of the rhombus:
\[ \text{Ratio} = \frac{\text{Area of Square}}{\text{Area of Rhombus}} = \frac{A^2}{\frac{A^2}{2}} = \frac{A^2 \cdot 2}{A^2} = 2 \]

### Conclusion
The ratio of the area of the square to the area of the rhombus is:
\[ \text{Ratio} = 2:1 \]