The sides of a parallelogram are 12 cm and 8 cm long and one of the diagonals is 10 cm long. If d is the length of other diagonal, then which one of the following is correct?

Options

A. `d lt 8 cm`
B. `8 cm lt d lt 10 cm`
C. `10 cm lt d lt 12 cm`
D. `d gt 12 cm`

Answer

To solve the problem step by step, we will use the properties of a parallelogram and the relationship between its sides and diagonals.

### Step 1: Understand the properties of a parallelogram
In a parallelogram, the sum of the squares of the diagonals is equal to the sum of the squares of the sides multiplied by 2. This can be expressed as:
\[ d_1^2 + d_2^2 = 2(a^2 + b^2) \]
where \( d_1 \) and \( d_2 \) are the lengths of the diagonals, and \( a \) and \( b \) are the lengths of the sides.

### Step 2: Identify the given values
From the problem, we know:
- Side \( a = 12 \) cm
- Side \( b = 8 \) cm
- One diagonal \( d_1 = 10 \) cm
- The other diagonal \( d_2 = d \) (unknown)

### Step 3: Substitute the known values into the formula
Using the formula from Step 1, we substitute the known values:
\[ 10^2 + d^2 = 2(12^2 + 8^2) \]

### Step 4: Calculate the squares of the sides
Now we calculate:
- \( 10^2 = 100 \)
- \( 12^2 = 144 \)
- \( 8^2 = 64 \)

### Step 5: Substitute the squares into the equation
Now substituting these values into the equation gives:
\[ 100 + d^2 = 2(144 + 64) \]

### Step 6: Calculate the right side of the equation
Calculate \( 144 + 64 = 208 \), then multiply by 2:
\[ 2 \times 208 = 416 \]

### Step 7: Set up the equation
Now we have:
\[ 100 + d^2 = 416 \]

### Step 8: Solve for \( d^2 \)
Subtract 100 from both sides:
\[ d^2 = 416 - 100 \]
\[ d^2 = 316 \]

### Step 9: Take the square root to find \( d \)
Now take the square root of both sides to find \( d \):
\[ d = \sqrt{316} \]
Calculating this gives:
\[ d \approx 17.76 \text{ cm} \]

### Step 10: Compare \( d \) with the given diagonal
Since we know one diagonal is 10 cm, we can conclude:
- \( d \approx 17.76 \text{ cm} \) is greater than 10 cm.

### Conclusion
The correct statement is that the length of the other diagonal \( d \) is greater than 10 cm.