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

Question 19

ABCDEF is a regular hexagon with sides of length $x$. This hexagon is enlarged, centre $F$, by scale factor $p$ to give hexagon $FGHIUK$. Show that the area of the s... show full transcript

Worked Solution & Example Answer:ABCDEF is a regular hexagon with sides of length $x$ - Edexcel - GCSE Maths - Question 19 - 2020 - Paper 3

Step 1

Find area of ABCDEF

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Answer

The area of a regular hexagon can be calculated with the formula:

$\text{Area} = \frac{3\sqrt{3}}{2} s^2$

where $s$ is the length of the sides. For hexagon ABCDEF, since the side length is $x$ , the area becomes:

$\text{Area}_{ABCDEF} = \frac{3\sqrt{3}}{2} x^2.$

Step 2

Find area of FGHIUK

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Answer

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:

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

Step 3

Calculate the area of the shaded region

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Answer

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:

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

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Questions answered

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

Worked Solution & Example Answer:ABCDEF is a regular hexagon with sides of length $x$ - Edexcel - GCSE Maths - Question 19 - 2020 - Paper 3

Find area of ABCDEF

Find area of FGHIUK

Calculate the area of the shaded region

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