To determine the length of the shorter side of a rectangle given that the sides are in a 3:4 ratio and the perimeter is 42 cm, follow these steps:
1. Denote the lengths of the sides as [tex]\(3x\)[/tex] and [tex]\(4x\)[/tex], where [tex]\(x\)[/tex] is a common multiplier.
2. The formula for the perimeter of a rectangle is given by:
[tex]\[ 2 \times (\text{length} + \text{width}) = \text{perimeter} \][/tex]
3. Substitute the sides of the rectangle ([tex]\(3x\)[/tex] and [tex]\(4x\)[/tex]) into the perimeter formula:
[tex]\[ 2 \times (3x + 4x) = 42 \][/tex]
4. Simplify the expression inside the parentheses:
[tex]\[ 2 \times 7x = 42 \][/tex]
5. Divide both sides by 2 to isolate [tex]\(7x\)[/tex]:
[tex]\[ 7x = 21 \][/tex]
6. Solve for [tex]\(x\)[/tex] by dividing both sides by 7:
[tex]\[ x = 3 \][/tex]
7. Now, calculate the lengths of the sides. The shorter side (width) is [tex]\(3x\)[/tex]:
[tex]\[ 3 \times 3 = 9 \][/tex]
Therefore, the length of the shorter side of the rectangle is [tex]\(9\)[/tex] cm.
