Given:
Perimeter of triangle = 120 units
Base of triangle = 20 units

To find the height of the triangle, we can use the formula for the perimeter of a triangle:
Perimeter = sum of all sides of the triangle

Since the base of the triangle is given as 20 units, we need to find the lengths of the other two sides. Let's assume the lengths of these sides as a and b.

Step 1: Assign variables to unknown sides:
Let the lengths of the two unknown sides be a and b.

Step 2: Write the perimeter equation:
Perimeter = a + b + base

Since the perimeter is given as 120 units and the base is given as 20 units, we can write the equation as:
120 = a + b + 20

Step 3: Rearrange the equation:
To isolate the terms with 'a' and 'b', we need to subtract 20 from both sides of the equation:
120 - 20 = a + b

Simplifying the equation, we get:
100 = a + b

Step 4: Determine the relationship between the sides:
In a triangle, the sum of the lengths of any two sides is always greater than the length of the third side.

We know that the base of the triangle is 20 units. So, the sum of the lengths of the other two sides (a and b) should be greater than 20 units.

Step 5: Find the range of values for a and b:
Since a + b > 20, we can say that a and b should be greater than 0 and less than 100.

Step 6: Determine the height of the triangle:
The height of a triangle can be calculated using the formula:
Area = (base * height) / 2

Since the base of the triangle is given as 20 units, we can substitute the value of the base in the formula:
Area = (20 * height) / 2

We can simplify the equation as:
Area = 10 * height

Step 7: Calculate the area of the triangle:
The area of a triangle can be calculated using the formula:
Area = sqrt(s * (s - a) * (s - b) * (s - c))

where s is the semi-perimeter of the triangle, given by:
s = (a + b + c) / 2

In this case, we have the perimeter of the triangle as 120 units. So, the semi-perimeter will be:
s = 120 / 2
s = 60

Now we can substitute the values of s, a, and b in the area formula:
Area = sqrt(60 * (60 - a) * (60 - b) * (60 - 20))

Step 8: Solve for height:
Since the area of a triangle is given by the formula Area = 10 * height, we can equate the two equations and solve for height:
10 * height = sqrt(60 * (60 - a) * (60 - b) * (60 - 20))

Simplifying the equation, we get:
height = sqrt((60 * (60 - a) * (60 - b) * (