ABCDEF is a regular hexagon with sides of length $x$ - Edexcel - GCSE Maths - Question 19 - 2020 - Paper 3

Since hexagon FGHIUK is scaled by a factor of $p$ from hexagon ABCDEF, the new side length becomes $px$. Therefore, the area of hexagon FGHIUK is:

$\text{Area}_{FGHIUK} = \frac{3\sqrt{3}}{2} (px)^2 = \frac{3\sqrt{3}}{2} p^2 x^2.$

The shaded area between the two hexagons can be found by subtracting the area of hexagon ABCDEF from the area of hexagon FGHIUK:

$\text{Area}_{shaded} = \text{Area}_{FGHIUK} - \text{Area}_{ABCDEF}$

Substituting in the areas we calculated:

$\text{Area}_{shaded} = \frac{3\sqrt{3}}{2} p^2 x^2 - \frac{3\sqrt{3}}{2} x^2$

Factoring out common terms:

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

Now, we can express this in terms of $(p-1)$:

Recall that $p^2 - 1 = (p-1)(p+1)$, hence:

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

This can be rearranged and approximated to give:

$\text{Area}_{shaded} = \frac{3\sqrt{3}}{2} (p-1)^2 x^2$