The diagram shows a regular polygon with 9 sides. Each side has length \((x - 2)\) cm. (a) Write an expression, in terms of \(x\), for the perimeter of this...

A regular polygon has equal sides, so perimeter \(=\) number of sides \(\times\) length of one side. This question uses that single relationship four times.

(a) With \(9\) sides each of length \((x - 2)\) cm:

\[ P = 9(x - 2) = 9x - 18 \ \text{cm} \] [B1]

The \(9\) multiplies both terms, so the constant becomes \(-18\), not \(-2\).

(b) Set the expression equal to the perimeter given:

\[ 9x - 18 = 72 \] [M1]

\[ 9x = 90 \quad \Rightarrow \quad x = 10 \] [A1]

Dividing first is an equally valid method: one side is \(72 \div 9 = 8\) cm, so \(x - 2 = 8\) and \(x = 10\).

(c) The side length is \((x - 2)\) cm, so

\[ 10 - 2 = 8 \ \text{cm} \] [B1]

(d) The new polygon is regular with the same side length of \(8\) cm and a perimeter of \(96\) cm. Dividing the perimeter by the side length gives the number of sides:

\[ \frac{96}{8} = 12 \] [B1]

So it has \(12\) sides. The reasoning in the last part is the same relationship read the other way round: knowing any two of perimeter, side length and number of sides determines the third. Note that \(x\) itself, \(10\), is not a side length and must not be used in that division.