Answer :

i) A regular hexagon is made up of 6 equilateral triangles.

Therefore, the green coloured triangles inside the yellow hexagon are equilateral triangles, that is all their sides are equal.

⇒ All the sides that comprises the hexagon are equal.

Hence, the inner red hexagon is also regular.

ii) Let the side of the inner red hexagon be x

⇒ the sides of the triangle will also be x

Let the side of outer yellow hexagon be y

Using Pythagoras theorem,

x^{2} + y^{2} = (2x)^{2}

⇒ y^{2} = (2x)^{2} – x^{2}

⇒ y^{2} = 4x^{2} – x^{2}

⇒ y^{2} = 3x^{2}

⇒ y= x√3

Hence, the outer hexagon’s side is √3 times the inner hexagon’s side.

iii) Now

Area of larger hexagon

⇒ Area of larger hexagon

And,

Area of smaller hexagon

Hence, the outer hexagon’s area is 3 times the inner hexagon’s area.

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