The perimeter of a triange with sides a,b,and c is 24 cm. Side a is 2cm longer than side b. The ration of the lenghts of sides b and c is 3:5. What are the lenghts of the three sides of the triangle?

a=b+2

5b=3c

a+b+c=24
(b+2)+b+(5/3 b) = 24
11/3 b = 22
b = 6

so, the sides are 8,6,10

To solve this problem, we can set up a system of equations.

Let's assign variables to the lengths of the sides:
Let b be the length of side b,
a be the length of side a, and
c be the length of side c.

From the information given, we can set up the following equations:

1. The perimeter of a triangle is the sum of its three sides, so we have:
a + b + c = 24

2. Side a is 2 cm longer than side b, so we have:
a = b + 2

3. The ratio of the lengths of sides b and c is 3:5, so we have:
b/c = 3/5

Now we can solve this system of equations to find the lengths of the sides.

From equation 2, we can substitute the value of "a" in equation 1:
(b + 2) + b + c = 24

Expanding equation 1:
2b + c = 22 (Equation 4)

From equation 3, we can rewrite it as:
b = (3/5) * c

Substituting this value in equation 4:
2((3/5) * c) + c = 22
(6/5) * c + c = 22
(11/5) * c = 22

Cross-multiplying:
11c = 110

Dividing both sides by 11:
c = 10

Substituting this value of "c" in equation 3:
b = (3/5) * 10
b = 6

Substituting the values of "b" and "c" in equation 1:
a + 6 + 10 = 24
a + 16 = 24
a = 8

Therefore, the lengths of the three sides of the triangle are:
Side a = 8 cm
Side b = 6 cm
Side c = 10 cm