ABCD is a parallelogram DP is perpendicular to AB. Given `BD = 5 cm, AP = 3 cm. and ` area of `Delta ADP = 6` sq. cm, find the perimeter of ABCD.

Answer

To find the perimeter of the parallelogram ABCD, we can follow these steps:

### Step 1: Understand the given information
We have:
- \( BD = 5 \, \text{cm} \)
- \( AP = 3 \, \text{cm} \)
- Area of triangle \( ADP = 6 \, \text{cm}^2 \)

### Step 2: Use the area of triangle formula
The area of triangle \( ADP \) can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is \( AP \) and the height is \( DP \). Therefore, we can write:
\[ 6 = \frac{1}{2} \times AP \times DP \]
Substituting \( AP = 3 \):
\[ 6 = \frac{1}{2} \times 3 \times DP \]

### Step 3: Solve for \( DP \)
Multiply both sides by 2:
\[ 12 = 3 \times DP \]
Now, divide by 3:
\[ DP = \frac{12}{3} = 4 \, \text{cm} \]

### Step 4: Use the Pythagorean theorem to find \( BP \)
In triangle \( BDP \), we can apply the Pythagorean theorem:
\[ BD^2 = BP^2 + DP^2 \]
Substituting the known values:
\[ 5^2 = BP^2 + 4^2 \]
This simplifies to:
\[ 25 = BP^2 + 16 \]
Now, subtract 16 from both sides:
\[ BP^2 = 25 - 16 = 9 \]
Taking the square root:
\[ BP = \sqrt{9} = 3 \, \text{cm} \]

### Step 5: Find the length of \( AD \)
In triangle \( ADP \), we again use the Pythagorean theorem:
\[ AD^2 = AP^2 + DP^2 \]
Substituting the known values:
\[ AD^2 = 3^2 + 4^2 \]
This simplifies to:
\[ AD^2 = 9 + 16 = 25 \]
Taking the square root:
\[ AD = \sqrt{25} = 5 \, \text{cm} \]

### Step 6: Find the lengths of the sides of the parallelogram
Since \( AB \) and \( DC \) are opposite sides of the parallelogram, we have:
\[ AB = AP + BP = 3 + 3 = 6 \, \text{cm} \]
And since \( AD \) and \( BC \) are also opposite sides:
\[ AD = 5 \, \text{cm} \]

### Step 7: Calculate the perimeter of the parallelogram
The perimeter \( P \) of parallelogram \( ABCD \) is given by:
\[ P = AB + BC + CD + DA \]
Since \( AB = DC = 6 \, \text{cm} \) and \( AD = BC = 5 \, \text{cm} \):
\[ P = 6 + 6 + 5 + 5 = 22 \, \text{cm} \]

### Final Answer
The perimeter of parallelogram ABCD is \( 22 \, \text{cm} \).

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