One side of the triangle is 2cm longer than the second side. The third is 3/5 as long as the first side. If the perimeter of the triangle is 24cm, Find the lengths of the three sides.

first side --- x

second side --- x-2
third side -- (3/5)x

x + x-2 + (3/5)x = 24
times 5

5x + 5x-10 + 3x = 120
13x = 130
x = 10

first side = 10
2nd side = 8
3rd side = 6

To solve this problem, let's assign variables to the lengths of the sides of the triangle.

Let's say the length of the second side is x cm.
Since "one side of the triangle is 2 cm longer than the second side," the length of the first side would be (x + 2) cm.
And since "the third side is 3/5 as long as the first side," the length of the third side would be (3/5)(x + 2) cm.

Now, we can use the given information that the perimeter of the triangle is 24 cm.

Perimeter = first side + second side + third side

24 = (x + 2) + x + (3/5)(x + 2)

To solve this equation, we can simplify it by multiplying through by 5 to eliminate the fraction:

120 = 5(x + 2) + 5x + 3(x + 2)
120 = 5x + 10 + 5x + 3x + 6
120 = 13x + 16
104 = 13x
x = 8

Now that we have the value of x, we can substitute it back into the expressions we set up for the lengths of the sides:

The length of the second side = x = 8 cm
The length of the first side = x + 2 = 8 + 2 = 10 cm
The length of the third side = (3/5)(x + 2) = (3/5)(8 + 2) = 6 cm

So, the lengths of the three sides of the triangle are 10 cm, 8 cm, and 6 cm, respectively.