Finding the Shortest Side of a Triangle Using Perimeter

This problem involves finding the length of the shortest side of a triangle when we know the relationships between its sides and its total perimeter.

Understanding the Triangle's Sides

Let's denote the length of the shortest side of the triangle as s.

• The length of the largest side is 3 times the shortest side, so its length is $3s$.
• The length of the third side is 2 cm more than the shortest side, so its length is $s + 2$ cm.

Using the Perimeter Information

The perimeter of a triangle is the sum of the lengths of all its sides.

We are given that the perimeter is 27 cm.

Therefore, we can write the equation:

Shortest side + Third side + Largest side = Perimeter

Substituting the expressions for the sides:

Step-by-Step Calculation

1. Combine like terms: Add all the terms involving s. $ (s + s + 3s) + 2 = 27 $ $ 5s + 2 = 27 $
2. Isolate the term with s: Subtract 2 from both sides of the equation. $ 5s = 27 - 2 $ $ 5s = 25 $
3. Solve for s: Divide both sides by 5 to find the length of the shortest side. $ s = \frac{25}{5} $ $ s = 5 $

Verifying the Solution

If the shortest side ($s$) is 5 cm:

• The third side is $s + 2 = 5 + 2 = 7$ cm.
• The largest side is $3s = 3 \times 5 = 15$ cm.

The perimeter is $5 \text{ cm} + 7 \text{ cm} + 15 \text{ cm} = 27 \text{ cm}$. This matches the given perimeter.

Conclusion

The length of the shortest side of the triangle is 5 cm.