A square and rhombus have the same base . If the rhombus is inclined at ` 60^(@)`, find the ratio of area of square to the area of the rhombus :

Options

A. `2 sqrt(3) : 3`
B. `1 : sqrt(3) `
C. `sqrt(3):2`
D. none of these

Answer

To find the ratio of the area of a square to the area of a rhombus with the same base and the rhombus inclined at 60 degrees, we can follow these steps:

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

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

### Step 3: Calculate the area of the rhombus
The area of a rhombus can be calculated using the formula:
\[ \text{Area of Rhombus} = \frac{1}{2} \times \text{base} \times \text{height} \]
However, since we are given the inclination angle, we can also use the formula:
\[ \text{Area of Rhombus} = \text{side}^2 \times \sin(\theta) \]
Given that all sides of the rhombus are equal to \( x \) and the angle \( \theta = 60^\circ \):
\[ \text{Area of Rhombus} = x^2 \times \sin(60^\circ) \]
We know that \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \), so:
\[ \text{Area of Rhombus} = x^2 \times \frac{\sqrt{3}}{2} = \frac{x^2 \sqrt{3}}{2} \]

### 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:
\[ \text{Ratio} = \frac{\text{Area of Square}}{\text{Area of Rhombus}} = \frac{x^2}{\frac{x^2 \sqrt{3}}{2}} = \frac{x^2 \times 2}{x^2 \sqrt{3}} = \frac{2}{\sqrt{3}} \]

### Step 5: Simplify the ratio
To express this ratio in a more standard form, we can multiply the numerator and the denominator by \( \sqrt{3} \):
\[ \text{Ratio} = \frac{2 \sqrt{3}}{3} \]

### Conclusion
Thus, the ratio of the area of the square to the area of the rhombus is:
\[ \text{Ratio} = 2\sqrt{3} : 3 \]