One of the diagonals of a rhombus is 70% of the other diagonal. What is the ratio of area of rhombus to the square of the length of the larger diagonal?

Options

A. `3: 10`
B. `3: 20`
C. `7:20`
D. `7: 10`

Answer

To solve the problem step by step, we will follow the given information about the rhombus and its diagonals.

### Step 1: Define the diagonals
Let the length of the larger diagonal be \( X \). According to the problem, the smaller diagonal is 70% of the larger diagonal. Therefore, we can express the smaller diagonal as:
\[ \text{Smaller diagonal} = 0.7X = \frac{7X}{10} \]

### Step 2: Calculate the area of the rhombus
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
where \( d_1 \) and \( d_2 \) are the lengths of the diagonals. Substituting the values we have:
\[ A = \frac{1}{2} \times X \times \frac{7X}{10} \]
\[ A = \frac{1}{2} \times \frac{7X^2}{10} \]
\[ A = \frac{7X^2}{20} \]

### Step 3: Calculate the square of the length of the larger diagonal
The square of the length of the larger diagonal is:
\[ X^2 \]

### Step 4: Find the ratio of the area of the rhombus to the square of the length of the larger diagonal
Now, we need to find the ratio of the area of the rhombus to the square of the length of the larger diagonal:
\[ \text{Ratio} = \frac{\text{Area of rhombus}}{\text{Square of the larger diagonal}} = \frac{\frac{7X^2}{20}}{X^2} \]
\[ \text{Ratio} = \frac{7X^2}{20X^2} = \frac{7}{20} \]

### Conclusion
The ratio of the area of the rhombus to the square of the length of the larger diagonal is:
\[ \frac{7}{20} \]