The longest side of a triangle is three times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is at least 61 cm, find the minimum length of the shortest side.

Explanation

To solve the problem, we will follow these steps:

1. **Define the variables**:
Let the shortest side of the triangle be \( x \).

2. **Express the other sides in terms of \( x \)**:
- The longest side is three times the shortest side, so it can be expressed as \( 3x \).
- The third side is 2 cm shorter than the longest side, so it can be expressed as \( 3x - 2 \).

3. **Write the perimeter inequality**:
The perimeter \( P \) of the triangle is the sum of all its sides. According to the problem, the perimeter is at least 61 cm. Therefore, we can write the inequality:
\[ x + 3x + (3x - 2) \geq 61 \]

4. **Simplify the inequality**:
Combine the terms on the left side:
\[ x + 3x + 3x - 2 = 7x - 2 \]
So, the inequality becomes:
\[ 7x - 2 \geq 61 \]

5. **Solve for \( x \)**:
Add 2 to both sides:
\[ 7x \geq 63 \]
Now, divide both sides by 7:
\[ x \geq 9 \]

6. **Conclusion**:
The minimum length of the shortest side \( x \) is 9 cm.

### Summary of the Steps:
1. Define the shortest side as \( x \).
2. Longest side: \( 3x \); Third side: \( 3x - 2 \).
3. Set up the perimeter inequality: \( 7x - 2 \geq 61 \).
4. Simplify to \( 7x \geq 63 \).
5. Solve for \( x \): \( x \geq 9 \).
6. Minimum length of the shortest side is 9 cm.