.4 The perimeter of a rectangle is \( 20 \mathrm{~cm}^{2} \). The one side of the rectangle is \( x \mathrm{~cm} \) and the other side is 2 cm shorter than the first side. Find the lengths of the two sides.

To find the lengths of the two sides of the rectangle, we can use the formula for the perimeter of a rectangle, which is given by \( P = 2(l + w) \), where \( l \) is the length and \( w \) is the width. Here, the perimeter is \( 20 \, \text{cm} \). Let’s denote the length as \( x \, \text{cm} \) and the width as \( w \, \text{cm} \). According to the problem, the width is 2 cm shorter than the length, so we can say \( w = x - 2 \). Now, putting these into the perimeter formula: \[ 2(x + (x - 2)) = 20 \] Simplifying that gives: \[ 2(2x - 2) = 20 \] \[ 4x - 4 = 20 \] \[ 4x = 24 \] \[ x = 6 \] Now substitute \( x \) back to find the width: \[ w = 6 - 2 = 4 \] So the lengths of the two sides are \( 6 \, \text{cm} \) and \( 4 \, \text{cm} \).