Find the longest side from a perimeter equation

Expert intro: This is one of those questions where the perimeter equation already does most of the work. Once you set it up, the longest side falls out quickly.

Detailed walkthrough

We are told the perimeter is 55 cm, so the side lengths must add to 55:

$x + (x+2) + (x+4) + (x+3) = 55$

Combine like terms:

$4x + 9 = 55$

Subtract 9 from both sides:

$4x = 46$

Divide by 4:

$x = 11.5$

Now find each side:

• $x = 11.5$
• $x+2 = 13.5$
• $x+3 = 14.5$
• $x+4 = 15.5$

So the longest side is 15.5 cm.

💡 Pitfall guide

A common mistake is to stop after finding $x$ and forget to check which expression is the largest. Another one is dropping one of the constants when you combine the terms: $2+4+3=9$, not 8.

🔄 Real-world variant

If the perimeter were changed to $P$ cm instead of 55, the same setup would give

$4x + 9 = P$

so

$x = \frac{P-9}{4}$

and the longest side would be

$x+4 = \frac{P+7}{4}$

That pattern is handy if the numbers in the shape change but the side expressions stay the same.

🔍 Related terms

perimeter, linear equation, algebraic expressions