If the sum of the diagonals of a rhombus is 12 cm, and its perimeter is `8 sqrt(5)` cm, then the lengths of the diagonals are :

Options

A. 6 cm and 6 cm
B. 7 cm and 5 cm
C. 8 cm and 4 cm
D. 9 cm and 3 cm

Answer

To solve the problem, we need to find the lengths of the diagonals of a rhombus given that the sum of the diagonals is 12 cm and the perimeter is \(8\sqrt{5}\) cm.

### Step-by-Step Solution:

1. **Understanding the Properties of a Rhombus**:
- Let the lengths of the diagonals be \(d_1\) and \(d_2\).
- According to the problem, we know that:
\[ d_1 + d_2 = 12 \quad \text{(Equation 1)} \]

2. **Using the Perimeter of the Rhombus**:
- The perimeter \(P\) of a rhombus can be expressed in terms of its diagonals as:
\[ P = 4 \times \text{side length} \]
- The side length \(s\) can also be calculated using the diagonals:
\[ s = \frac{1}{2} \sqrt{d_1^2 + d_2^2} \]
- Therefore, the perimeter can be rewritten as:
\[ P = 4 \times \frac{1}{2} \sqrt{d_1^2 + d_2^2} = 2\sqrt{d_1^2 + d_2^2} \]
- Given that the perimeter is \(8\sqrt{5}\), we have:
\[ 2\sqrt{d_1^2 + d_2^2} = 8\sqrt{5} \]
- Dividing both sides by 2:
\[ \sqrt{d_1^2 + d_2^2} = 4\sqrt{5} \]
- Squaring both sides gives us:
\[ d_1^2 + d_2^2 = (4\sqrt{5})^2 = 16 \times 5 = 80 \quad \text{(Equation 2)} \]

3. **Substituting Equation 1 into Equation 2**:
- From Equation 1, we can express \(d_2\) in terms of \(d_1\):
\[ d_2 = 12 - d_1 \]
- Substituting this into Equation 2:
\[ d_1^2 + (12 - d_1)^2 = 80 \]
- Expanding the equation:
\[ d_1^2 + (144 - 24d_1 + d_1^2) = 80 \]
- Combining like terms:
\[ 2d_1^2 - 24d_1 + 144 = 80 \]
- Simplifying:
\[ 2d_1^2 - 24d_1 + 64 = 0 \]
- Dividing the entire equation by 2:
\[ d_1^2 - 12d_1 + 32 = 0 \quad \text{(Quadratic Equation)} \]

4. **Solving the Quadratic Equation**:
- We can use the quadratic formula \(d_1 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
- Here, \(a = 1\), \(b = -12\), and \(c = 32\).
- Calculating the discriminant:
\[ b^2 - 4ac = (-12)^2 - 4 \times 1 \times 32 = 144 - 128 = 16 \]
- Now substituting into the quadratic formula:
\[ d_1 = \frac{12 \pm \sqrt{16}}{2} = \frac{12 \pm 4}{2} \]
- This gives us two possible values:
\[ d_1 = \frac{16}{2} = 8 \quad \text{and} \quad d_1 = \frac{8}{2} = 4 \]

5. **Finding the Lengths of the Diagonals**:
- If \(d_1 = 8\), then:
\[ d_2 = 12 - 8 = 4 \]
- If \(d_1 = 4\), then:
\[ d_2 = 12 - 4 = 8 \]
- Thus, the lengths of the diagonals are \(8 \, \text{cm}\) and \(4 \, \text{cm}\).

### Final Answer:
The lengths of the diagonals are \(8 \, \text{cm}\) and \(4 \, \text{cm}\).