Given:
Perimeter of the right angle triangle = 72 cm
Area of the right angle triangle = 216 cm²

To find:
Sum of the lengths of the perpendicular sides of the triangle

Solution:

Step 1: Identify the Perpendicular Sides
In a right angle triangle, the perpendicular sides are the two sides that form the right angle. Let's assume these sides to be a and b.

Step 2: Use Perimeter Formula
The perimeter of a triangle is the sum of all its sides. In this case, the perimeter is given as 72 cm. So we can write the equation as:
a + b + hypotenuse = 72 cm

Step 3: Use Area Formula
The area of a triangle can be calculated using the formula:
Area = (base * height) / 2
Since it's a right angle triangle, one of the perpendicular sides can be considered as the base, and the other perpendicular side as the height. So we have:
(1/2) * a * b = 216 cm²

Step 4: Simplify the Equations
From the perimeter equation, we can rewrite it as:
hypotenuse = 72 cm - a - b

Substituting this value of hypotenuse in the area equation, we get:
(1/2) * a * b = 216 cm²

Step 5: Solve the Equations
We can rearrange the area equation to isolate one variable in terms of the other:
a = (432 cm²) / b

Substituting this value of a in the perimeter equation:
hypotenuse = 72 cm - (432 cm² / b) - b

Step 6: Simplify the Equations Further
To find the sum of the lengths of the perpendicular sides (a + b), we need to eliminate the hypotenuse. We can square both sides of the perimeter equation to get rid of the square root:
hypotenuse² = (72 cm - (432 cm² / b) - b)²

Step 7: Substitute the Hypotenuse in the Area Equation
Now, substitute the value of hypotenuse from the squared perimeter equation into the area equation:
(1/2) * a * b = 216 cm²
(1/2) * a * b = 216 cm²

Step 8: Solve the Quadratic Equation
By simplifying and solving the quadratic equation, we can find the possible values of b. Once we have the values of b, we can substitute them back into the perimeter equation to find the corresponding values of a. Finally, we can calculate the sum of the lengths of the perpendicular sides (a + b) for each set of solutions.

Step 9: Evaluate the Sum of Perpendicular Sides
Calculate the sum of the lengths of the perpendicular sides (a + b) using the obtained values of a and b from the previous step.

Step 10: Provide the Answer
The sum of the lengths of the perpendicular sides of the right angle triangle is the sum of (a + b) calculated in the previous step.

Answer will be 42.