Area of a rhombus is ` 144 cm^(2)` and the ratio of length of two diagonals is 1:2. The sum of lengths of its diagonals are :

Options

A. 72 cm
B. 40 cm
C. 36 cm
D. `18 sqrt(2) cm `

Answer

To find the sum of the lengths of the diagonals of a rhombus given its area and the ratio of the diagonals, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- Area of the rhombus = 144 cm²
- Ratio of lengths of the diagonals (d1 : d2) = 1 : 2

2. **Assign Variables**:
- Let the length of the first diagonal (d1) be \( x \).
- Then, the length of the second diagonal (d2) can be expressed as \( 2x \) (since the ratio is 1:2).

3. **Use the Area Formula**:
- The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d1 \times d2 \]
- Substituting the values of d1 and d2:
\[ 144 = \frac{1}{2} \times x \times 2x \]

4. **Simplify the Equation**:
- The equation simplifies to:
\[ 144 = \frac{1}{2} \times 2x^2 \]
- This further simplifies to:
\[ 144 = x^2 \]

5. **Solve for x**:
- Taking the square root of both sides:
\[ x = \sqrt{144} = 12 \text{ cm} \]

6. **Find the Lengths of the Diagonals**:
- Now, calculate the lengths of the diagonals:
- d1 = \( x = 12 \) cm
- d2 = \( 2x = 2 \times 12 = 24 \) cm

7. **Calculate the Sum of the Diagonals**:
- The sum of the lengths of the diagonals is:
\[ d1 + d2 = 12 + 24 = 36 \text{ cm} \]

### Final Answer:
The sum of the lengths of the diagonals is **36 cm**.

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