ABCDEF is a regular hexagon with sides of length $x$ - Edexcel - GCSE Maths - Question 20 - 2020 - Paper 3

Since $FGHUK$ is an enlarged hexagon with a scale factor of $p$, the area can be computed as:

$A_{FGHUK} = \frac{3\sqrt{3}}{2}(px)^2$

which simplifies to:

$A_{FGHUK} = \frac{3\sqrt{3}}{2}p^2x^2.$

The area of the shaded region can be calculated by subtracting the area of $ABCDEF$ from the area of $FGHUK$:

$A_{shaded} = A_{FGHUK} - A_{ABCDEF}$

Substituting the areas we calculated previously gives:

$A_{shaded} = \frac{3\sqrt{3}}{2}p^2x^2 - \frac{3\sqrt{3}}{2}x^2$

Factoring out the common terms yields:

$A_{shaded} = \frac{3\sqrt{3}}{2}(p^2 - 1)x^2.$

Recognizing that $p^2 - 1 = (p-1)(p+1)$ completes the derivation.

We can rewrite the area of the shaded region as:

$A_{shaded} = \frac{3\sqrt{3}}{2}(p-1)(p+1)x^2.$

Since the problem specifies that we should demonstrate the area in terms of $p-1$, the final expression is:

$A_{shaded} = \frac{3\sqrt{3}}{2}(p-1)x^2$.