The problem states that quadrilateral $ABCD$ has a perimeter of 95 centimeters. The side lengths are given as $AB = 3a + 2$, $BC = 2(a-1)$, $CD = 6a + 4$, and $AD = 5a - 5$. We need to find the length of each side.

1. Problem Description

The problem states that quadrilateral $ABCD$ has a perimeter of 95 centimeters. The side lengths are given as $AB = 3a + 2$, $BC = 2(a-1)$, $CD = 6a + 4$, and $AD = 5a - 5$. We need to find the length of each side.

2. Solution Steps

The perimeter of a quadrilateral is the sum of its four sides. Thus,

$AB + BC + CD + AD = 95$.

Substituting the given expressions for the side lengths, we get:

$(3a + 2) + (2(a-1)) + (6a + 4) + (5a - 5) = 95$.

First, simplify the equation by expanding $2(a-1)$:

$3a + 2 + 2a - 2 + 6a + 4 + 5a - 5 = 95$.

Combine like terms:

$(3a + 2a + 6a + 5a) + (2 - 2 + 4 - 5) = 95$

$16a - 1 = 95$.

Add 1 to both sides:

$16a = 96$.

Divide both sides by 16:

$a = \frac{96}{16} = 6$.

Now, substitute $a = 6$ into the expressions for each side length:

$AB = 3a + 2 = 3(6) + 2 = 18 + 2 = 20$.

$BC = 2(a - 1) = 2(6 - 1) = 2(5) = 10$.

$CD = 6a + 4 = 6(6) + 4 = 36 + 4 = 40$.

$AD = 5a - 5 = 5(6) - 5 = 30 - 5 = 25$.

3. Final Answer

$AB = 20$ cm

$BC = 10$ cm

$CD = 40$ cm

$AD = 25$ cm