The sides of a triangle are proportional to the numbers 3, 4, and 5, and the perimeter is 72 cm. Find the smallest side of the triangle.

The sides of the triangle are proportional to the numbers 3, 4, and 5. Let the sides of the triangle be \(3x\), \(4x\), and \(5x\), where \(x\) is the common multiplier.

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

\[
3x + 4x + 5x = 72
\]

Simplifying:

\[
12x = 72
\]

Now, solve for \(x\):

\[
x = \frac{72}{12} = 6
\]

Thus, the lengths of the sides are:

- \(3x = 3 \times 6 = 18\) cm
- \(4x = 4 \times 6 = 24\) cm
- \(5x = 5 \times 6 = 30\) cm

The smallest side is \(18\) cm.

Would you like further details or have any questions?

Here are some related questions:
1. How do you determine the largest side of a triangle in proportional problems?
2. What is the area of this triangle using Heron's formula?
3. What is the importance of the perimeter in solving triangle side problems?
4. Can you generalize this problem for different sets of proportions?
5. How does the Pythagorean theorem apply to triangles with sides proportional to 3, 4, and 5?

**Tip:** In proportional problems, always set the sides equal to the ratio multiplied by a common variable.

Learn how to find the smallest side of a triangle when the sides are in proportion to 3, 4, and 5, and the perimeter is 72 cm. This problem involves understanding proportions, solving linear equations, and applying the basic proportionality theorem. Suitable for students in Grades 6-8.

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