If the sum of diagonals of a rhombus is 10 cm and its area is `12 cm^2`, then the lengths of its diagonals are :

Options

A. 5,5
B. 9,1
C. 8,2
D. 6,4

Answer

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

### Step-by-Step Solution:

1. **Define the Variables**:
Let the lengths of the diagonals be \( d_1 \) and \( d_2 \).

2. **Set Up the Equations**:
We know from the problem that:
- The sum of the diagonals:
\[ d_1 + d_2 = 10 \quad \text{(Equation 1)} \]
- The area of the rhombus can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 = 12 \quad \text{(Equation 2)} \]

3. **Rearrange Equation 1**:
From Equation 1, we can express \( d_1 \) in terms of \( d_2 \):
\[ d_1 = 10 - d_2 \quad \text{(Equation 3)} \]

4. **Substitute Equation 3 into Equation 2**:
Substitute \( d_1 \) from Equation 3 into Equation 2:
\[ \frac{1}{2} \times (10 - d_2) \times d_2 = 12 \]

5. **Multiply through by 2**:
To eliminate the fraction, multiply both sides by 2:
\[ (10 - d_2) \times d_2 = 24 \]

6. **Expand and Rearrange**:
Expanding the left side gives:
\[ 10d_2 - d_2^2 = 24 \]
Rearranging this equation gives:
\[ d_2^2 - 10d_2 + 24 = 0 \quad \text{(Equation 4)} \]

7. **Factor Equation 4**:
We need to factor the quadratic equation:
\[ (d_2 - 6)(d_2 - 4) = 0 \]
This gives us two possible solutions for \( d_2 \):
\[ d_2 = 6 \quad \text{or} \quad d_2 = 4 \]

8. **Find Corresponding Values of \( d_1 \)**:
Using Equation 3, we can find \( d_1 \):
- If \( d_2 = 6 \):
\[ d_1 = 10 - 6 = 4 \]
- If \( d_2 = 4 \):
\[ d_1 = 10 - 4 = 6 \]

9. **Conclusion**:
The lengths of the diagonals are \( d_1 = 4 \, \text{cm} \) and \( d_2 = 6 \, \text{cm} \) (or vice versa).

### Final Answer:
The lengths of the diagonals are \( 4 \, \text{cm} \) and \( 6 \, \text{cm} \).

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