Two different sides of a parallelogram are 8 cm and 6 cm and the ratio of the diagonals is 3 : 4. Find the difference between the lengths of the diagonals?

Options

A. 5 cm
B. 7 cm
C. 8 cm
D. `sqrt(8) cm`

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Define the diagonals
Let the lengths of the diagonals of the parallelogram be represented as:
- Diagonal 1 = 3x
- Diagonal 2 = 4x

### Step 2: Use the formula for diagonals in a parallelogram
The relationship between the sides and diagonals of a parallelogram is given by the formula:
\[ d_1^2 + d_2^2 = 2(a^2 + b^2) \]
where \(d_1\) and \(d_2\) are the lengths of the diagonals, and \(a\) and \(b\) are the lengths of the sides.

### Step 3: Substitute the values
Given that the sides of the parallelogram are 8 cm and 6 cm, we can substitute these values into the formula:
\[ (3x)^2 + (4x)^2 = 2(8^2 + 6^2) \]

### Step 4: Calculate the squares
Calculating the squares:
\[ (3x)^2 = 9x^2 \]
\[ (4x)^2 = 16x^2 \]
Thus, we have:
\[ 9x^2 + 16x^2 = 2(64 + 36) \]

### Step 5: Simplify the equation
Combine the left side:
\[ 25x^2 = 2(100) \]
This simplifies to:
\[ 25x^2 = 200 \]

### Step 6: Solve for x
Now, divide both sides by 25:
\[ x^2 = \frac{200}{25} = 8 \]
Taking the square root gives:
\[ x = \sqrt{8} = 2\sqrt{2} \]

### Step 7: Find the lengths of the diagonals
Now, we can find the lengths of the diagonals:
- Diagonal 1 = \(3x = 3(2\sqrt{2}) = 6\sqrt{2}\)
- Diagonal 2 = \(4x = 4(2\sqrt{2}) = 8\sqrt{2}\)

### Step 8: Calculate the difference between the diagonals
The difference between the lengths of the diagonals is:
\[ \text{Difference} = 8\sqrt{2} - 6\sqrt{2} = 2\sqrt{2} \]

### Final Answer
The difference between the lengths of the diagonals is \(2\sqrt{2}\) cm.

---