To find the longer side of the rectangle, we know the perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2(l + w) \]
where \( l \) is the length and \( w \) is the width. Given that the perimeter \( P = 64 \) cm, we can set up the equation:
\[ 64 = 2(l + w) \]
Dividing both sides by 2:
\[ 32 = l + w \]
From the information provided, we have:
- Side \( AB = 3x \) (width)
- Side \( AD = 5x \) (length)
Thus, we can express the perimeter equation in terms of \( x \):
\[ l = 5x \quad \text{and} \quad w = 3x \]
Substituting into the equation \( l + w = 32 \):
\[ 5x + 3x = 32 \]
Simplifying:
\[ 8x = 32 \]
Dividing by 8:
\[ x = 4 \]
Now, we can find the lengths of the sides:
- Length \( l = 5x = 5(4) = 20 \) cm
- Width \( w = 3x = 3(4) = 12 \) cm
The longer side of the rectangle is therefore:
\[ \boxed{20 \text{ cm}} \]
So the correct response is 20 cm.