We are given the following information about a triangle: - The longest side is 8 cm longer than side $a$. - Side $b$ is 12 cm less than twice the length of side $a$. - The perimeter of the triangle is 40 cm. We need to find the length of each side.

1. Problem Description

We are given the following information about a triangle:

- The longest side is 8 cm longer than side $a$.

- Side $b$ is 12 cm less than twice the length of side $a$.

- The perimeter of the triangle is 40 cm.

We need to find the length of each side.

2. Solution Steps

Let the sides of the triangle be $a$, $b$, and $c$, where $c$ is the longest side.

From the given information, we have:

$c = a + 8$

$b = 2a - 12$

The perimeter of the triangle is the sum of its sides:

$a + b + c = 40$

Substitute the expressions for $b$ and $c$ in terms of $a$ into the perimeter equation:

$a + (2a - 12) + (a + 8) = 40$

Combine like terms:

$4a - 4 = 40$

Add 4 to both sides:

$4a = 44$

Divide both sides by 4:

$a = 11$

Now we can find the lengths of the other sides:

$b = 2a - 12 = 2(11) - 12 = 22 - 12 = 10$

$c = a + 8 = 11 + 8 = 19$

Therefore, the lengths of the sides are $a=11$ cm, $b=10$ cm, and $c=19$ cm.

3. Final Answer

The length of side a is 11 cm, the length of side b is 10 cm, and the length of the longest side is 19 cm.