We are given that the perimeter of the triangle is 22 cm, and the expressions for the three sides of the triangle are:

• First side: $3x + 1$
• Second side: $2x - 1$
• Third side: $x + 4$

Step 1: Use the formula for the perimeter

The perimeter of a triangle is the sum of the lengths of all its sides. Therefore, we can write:

$(3x + 1) + (2x - 1) + (x + 4) = 22$

Step 2: Simplify the equation

Combine the like terms:

$3x + 2x + x + 1 - 1 + 4 = 22$

This simplifies to:

$6x + 4 = 22$

Step 3: Solve for $x$

Subtract 4 from both sides:

$6x = 18$

Now, divide by 6:

$x = 3$

Step 4: Find the lengths of the sides

Now that we know $x = 3$, substitute this value into the expressions for the sides:

• First side: $3(3) + 1 = 9 + 1 = 10$
• Second side: $2(3) - 1 = 6 - 1 = 5$
• Third side: $3 + 4 = 7$

Step 5: Identify the shortest side

The three sides of the triangle are 10 cm, 5 cm, and 7 cm. Therefore, the shortest side is 5 cm.

Would you like more details or explanations?

Here are 5 related questions you can explore:

1. How do you find the perimeter of a triangle when given the side lengths?
2. Can a triangle with sides 10 cm, 5 cm, and 7 cm exist? (Hint: triangle inequality)
3. How would the problem change if the perimeter was 30 cm instead of 22 cm?
4. What if the expressions for the sides were in terms of different variables?
5. Can you generalize this process for any polygon?

Tip: Always check the triangle inequality to ensure the side lengths form a valid triangle!