The sides of a triangle are consecutive integers. The perimeter of the triangle is 120 cm. Find the length of the greatest side.

### Concept & Consecutive Integers Consecutive integers can be represented algebraically as $x$, $x+1$, and $x+2$. However, defining them symmetrically as $x-1$, $x$, and $x+1$ is a powerful algebraic trick because their sum perfectly cancels out the integer constants. ### Step-by-Step Solution * **Given:** The three sides are strictly consecutive integers. The perimeter (the sum of all sides) is exactly 120 cm. * **Set up Variables:** Let the three sides be represented symmetrically as $(x - 1)$, $x$, and $(x + 1)$. * **Formulate the Equation:** $(x - 1) + x + (x + 1) = 120$ * **Solve for $x$:** Notice how the $-1$ and $+1$ cancel out effortlessly. $3x = 120$ $x = 40$ * **Find Greatest Side:** The individual sides are $40-1=39$, $40$, and $40+1=41$. The greatest side is represented by $(x + 1) = 41\text{ cm}$. ### Exam Strategy & Shortcut If three consecutive numbers sum up to a specific value, the middle number is always exactly the average of that sum. $120 / 3 = 40$. So the consecutive numbers must be 39, 40, and 41. The greatest of these is clearly 41. This allows you to solve the entire problem mentally in just seconds. ### Common Pitfall Setting the sides up the traditional way as $x$, $x+1$, $x+2$ ($3x+3=120 \Rightarrow 3x=117 \Rightarrow x=39$), and then hastily selecting 39 (which is the smallest side) instead of completing the final required step to find the greatest side ($x+2 = 41$). ### Final Answer Therefore, the correct answer is **41 cm**.