A parallelogram PQRS ,the length of whose sides are 8 cm and 12 cm ,has one diagonal 10 cm long .The length of the other diagonal is approximately :

Options

A. 17.8 cm
B. 17.5 cm
C. 17 cm
D. 18 cm

Answer

To find the length of the other diagonal in the parallelogram PQRS, we can use the formula relating the lengths of the diagonals and the sides of the parallelogram:

\[ d_1^2 + d_2^2 = 2(l^2 + b^2) \]

Where:
- \( d_1 \) and \( d_2 \) are the lengths of the diagonals,
- \( l \) is the length of one side,
- \( b \) is the length of the other side.

Given:
- \( l = 12 \, \text{cm} \)
- \( b = 8 \, \text{cm} \)
- \( d_1 = 10 \, \text{cm} \)

We need to find \( d_2 \).

### Step 1: Calculate \( l^2 \) and \( b^2 \)
First, we calculate \( l^2 \) and \( b^2 \):
\[ l^2 = 12^2 = 144 \]
\[ b^2 = 8^2 = 64 \]

### Step 2: Calculate \( 2(l^2 + b^2) \)
Now, we calculate \( 2(l^2 + b^2) \):
\[ l^2 + b^2 = 144 + 64 = 208 \]
\[ 2(l^2 + b^2) = 2 \times 208 = 416 \]

### Step 3: Substitute values into the diagonal formula
Now we substitute \( d_1 \) and the calculated value into the formula:
\[ d_1^2 + d_2^2 = 416 \]
Substituting \( d_1 = 10 \):
\[ 10^2 + d_2^2 = 416 \]
\[ 100 + d_2^2 = 416 \]

### Step 4: Solve for \( d_2^2 \)
Now, we solve for \( d_2^2 \):
\[ d_2^2 = 416 - 100 \]
\[ d_2^2 = 316 \]

### Step 5: Calculate \( d_2 \)
Now, we take the square root to find \( d_2 \):
\[ d_2 = \sqrt{316} \]
To approximate \( \sqrt{316} \):
\[ \sqrt{316} \approx 17.78 \, \text{cm} \]

Thus, the length of the other diagonal \( d_2 \) is approximately \( 17.78 \, \text{cm} \).

### Final Answer
The length of the other diagonal is approximately \( 17.8 \, \text{cm} \).

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