One of the diagonal of a parallelogram is 18 cm, whose adjacent sides are 16 cm and 20 cm respectively. What is the length of other diagonal?

Options

A. `2sqrt(247)cm`
B. `13cm`
C. `15.2cm`
D. `28.5cm`

Answer

To find the length of the other diagonal of the parallelogram, we can use the property of the diagonals in a parallelogram. The formula relating the lengths of the diagonals and the sides is:

\[ D_1^2 + D_2^2 = 2(A^2 + B^2) \]

Where:
- \( D_1 \) and \( D_2 \) are the lengths of the diagonals,
- \( A \) and \( B \) are the lengths of the adjacent sides.

Given:
- \( D_1 = 18 \, \text{cm} \)
- \( A = 16 \, \text{cm} \)
- \( B = 20 \, \text{cm} \)

We need to find \( D_2 \).

### Step 1: Substitute the known values into the formula.

Using the formula:
\[ D_1^2 + D_2^2 = 2(A^2 + B^2) \]

Substituting the values we have:
\[ 18^2 + D_2^2 = 2(16^2 + 20^2) \]

### Step 2: Calculate \( 18^2 \), \( 16^2 \), and \( 20^2 \).

Calculating each square:
- \( 18^2 = 324 \)
- \( 16^2 = 256 \)
- \( 20^2 = 400 \)

### Step 3: Substitute the squared values back into the equation.

Now substituting back:
\[ 324 + D_2^2 = 2(256 + 400) \]

### Step 4: Calculate \( 256 + 400 \) and then multiply by 2.

Calculating:
\[ 256 + 400 = 656 \]
Then,
\[ 2 \times 656 = 1312 \]

### Step 5: Set up the equation to solve for \( D_2^2 \).

Now we have:
\[ 324 + D_2^2 = 1312 \]

### Step 6: Isolate \( D_2^2 \).

Subtract \( 324 \) from both sides:
\[ D_2^2 = 1312 - 324 \]
\[ D_2^2 = 988 \]

### Step 7: Take the square root to find \( D_2 \).

Now, taking the square root:
\[ D_2 = \sqrt{988} \]

### Step 8: Simplify \( \sqrt{988} \).

We can simplify \( \sqrt{988} \):
\[ \sqrt{988} = \sqrt{4 \times 247} = 2\sqrt{247} \]

### Final Answer:

Thus, the length of the other diagonal \( D_2 \) is:
\[ D_2 = 2\sqrt{247} \, \text{cm} \]

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