If x and y are respectively the areas of a square and a rhombus of sides of same length. Then what is `x:y` ?

Options

A. `1:1`
B. `2:sqrt(3)`
C. `4:sqrt(3)`
D. `3:2`

Answer

To solve the problem, we need to find the ratio of the areas of a square and a rhombus, both having sides of the same length. Let's denote the side length of both shapes as \( a \).

### Step-by-step Solution:

1. **Identify the Area of the Square**:
- The area \( x \) of a square is given by the formula:
\[ x = \text{side}^2 = a^2 \]

2. **Identify the Area of the Rhombus**:
- The area \( y \) of a rhombus can be calculated using the formula:
\[ y = \text{base} \times \text{height} \]
- In this case, the base of the rhombus is also \( a \). However, to find the height, we need to consider the properties of the rhombus. For a rhombus with all sides equal to \( a \), if we assume the angle between two sides is \( \theta \), the height can be expressed as:
\[ \text{height} = a \sin(\theta) \]
- Therefore, the area \( y \) of the rhombus becomes:
\[ y = a \times (a \sin(\theta)) = a^2 \sin(\theta) \]

3. **Finding the Ratio \( x:y \)**:
- Now, we need to find the ratio \( x:y \):
\[ \frac{x}{y} = \frac{a^2}{a^2 \sin(\theta)} = \frac{1}{\sin(\theta)} \]

4. **Determining the Special Case**:
- If the rhombus is a square (which is a special case of a rhombus), then \( \theta = 90^\circ \), and \( \sin(90^\circ) = 1 \). Thus, in this case:
\[ \frac{x}{y} = \frac{1}{1} = 1 \]

5. **Conclusion**:
- Therefore, the ratio \( x:y \) is:
\[ x:y = 1:1 \]

### Final Answer:
The ratio of the areas \( x:y \) is \( 1:1 \).