The area of a parallelogram PQRS is `162cm^(2)` This parallelogram has adjacent sides PQ = 27 cm and QR = 12 cm . Find the distance between its longer sides and that between its shorter sides .

Answer

To solve the problem step-by-step, we will use the formula for the area of a parallelogram, which is given by:

\[ \text{Area} = \text{Base} \times \text{Height} \]

### Step 1: Identify the given values
We are given:
- Area of the parallelogram \( = 162 \, \text{cm}^2 \)
- Length of side \( PQ = 27 \, \text{cm} \) (longer side)
- Length of side \( QR = 12 \, \text{cm} \) (shorter side)

### Step 2: Calculate the distance between the longer sides
To find the distance between the longer sides (which is the height corresponding to the base PQ), we use the area formula:

\[ \text{Area} = \text{Base} \times \text{Height} \]

Here, we take \( PQ \) as the base. Thus,

\[ 162 = 27 \times \text{Height}_{PQ} \]

Rearranging the formula to find the height:

\[ \text{Height}_{PQ} = \frac{162}{27} \]

Calculating the height:

\[ \text{Height}_{PQ} = 6 \, \text{cm} \]

### Step 3: Calculate the distance between the shorter sides
Now, we find the distance between the shorter sides (which is the height corresponding to the base QR). Again, using the area formula:

\[ \text{Area} = \text{Base} \times \text{Height} \]

Here, we take \( QR \) as the base. Thus,

\[ 162 = 12 \times \text{Height}_{QR} \]

Rearranging the formula to find the height:

\[ \text{Height}_{QR} = \frac{162}{12} \]

Calculating the height:

\[ \text{Height}_{QR} = 13.5 \, \text{cm} \]

### Final Answers
- The distance between the longer sides is \( 6 \, \text{cm} \).
- The distance between the shorter sides is \( 13.5 \, \text{cm} \).