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, we need to find the ratio of the area of a rhombus to the square of the length of the larger diagonal. Let's break it down step by step.

### Step 1: Define the diagonals
Let the length of the larger diagonal be \( d_1 \) and the length of the smaller diagonal be \( d_2 \). According to the problem, one diagonal is 70% of the other diagonal. We can express this relationship mathematically:
\[ d_2 = 0.7 \cdot d_1 \]

### Step 2: Express the area of the rhombus
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{d_1 \cdot d_2}{2} \]
Substituting \( d_2 \) from Step 1 into the area formula gives:
\[ A = \frac{d_1 \cdot (0.7 \cdot d_1)}{2} = \frac{0.7 \cdot d_1^2}{2} \]

### Step 3: Calculate the square of the larger diagonal
The square of the larger diagonal \( d_1 \) is:
\[ d_1^2 \]

### Step 4: Find the ratio of the area to the square of the larger diagonal
Now, we need to find the ratio of the area \( A \) to the square of the larger diagonal \( d_1^2 \):
\[ \text{Ratio} = \frac{A}{d_1^2} = \frac{\frac{0.7 \cdot d_1^2}{2}}{d_1^2} \]
This simplifies to:
\[ \text{Ratio} = \frac{0.7}{2} = 0.35 \]

### Step 5: Express the ratio in fractional form
The ratio \( 0.35 \) can be expressed as:
\[ \text{Ratio} = \frac{35}{100} = \frac{7}{20} \]

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

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