The ratio of the length of the diagonal of a rhombus is 2:5. Then, the ratio of the area of the rhombus to the square of the shorter diagonal :

Options

A. `5:4`
B. `5:2`
C. `2:5`
D. None of these

Answer

To solve the problem, we need to find the ratio of the area of a rhombus to the square of its shorter diagonal, given the ratio of the lengths of the diagonals.

### Step-by-Step Solution:

1. **Identify the diagonals**:
Let the lengths of the diagonals of the rhombus be \( d_1 \) and \( d_2 \). According to the problem, the ratio of the diagonals is given as \( 2:5 \). We can express this as:
\[ d_1 = 2x \quad \text{and} \quad d_2 = 5x \]

2. **Calculate the area of the rhombus**:
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
Substituting the values of \( d_1 \) and \( d_2 \):
\[ A = \frac{1}{2} \times (2x) \times (5x) = \frac{1}{2} \times 10x^2 = 5x^2 \]

3. **Calculate the square of the shorter diagonal**:
The shorter diagonal is \( d_1 = 2x \). Therefore, the square of the shorter diagonal is:
\[ (d_1)^2 = (2x)^2 = 4x^2 \]

4. **Find the ratio of the area of the rhombus to the square of the shorter diagonal**:
We need to find the ratio:
\[ \text{Ratio} = \frac{\text{Area of rhombus}}{(\text{Shorter diagonal})^2} = \frac{5x^2}{4x^2} \]
Simplifying this gives:
\[ \text{Ratio} = \frac{5}{4} \]

### Final Answer:
The ratio of the area of the rhombus to the square of the shorter diagonal is \( \frac{5}{4} \).