Q1.
Find the perimeter of squares with sides 1 cm, 1.5 cm, 2 cm, 2.5 cm and 3 cm. What will happen to the perimeters if the sides increase by 0.5 cm?
Answer
The perimeter of a square of side x is the linear polynomial 4x.
| Side x (cm) | 1 | 1.5 | 2 | 2.5 | 3 |
|---|---|---|---|---|---|
| Perimeter 4x (cm) | 4 | 6 | 8 | 10 | 12 |
| Increase | — | +2 | +2 | +2 | +2 |
Each extra 0.5 cm of side adds 2 cm to the perimeter, every single time.
Why it happens: do not just read it off the table — prove it. For any side x,
4(x + 0.5) − 4x = 4x + 2 − 4x = 2
The x cancels, so the increase does not depend on where you started. That cancellation is possible only because the variable appears to the first power; it is the algebraic reason a linear polynomial produces equal steps.
Check it yourself: try the same subtraction on a square’s area, x². (x + 0.5)² − x² = x + 0.25, which still contains x — so the area grows by different amounts at different stages. That is precisely what makes area non-linear.