Question : ABCDEF is a regular hexagon of side 12 cm. What is the area (in cm2) of the $\triangle $ECD?
Option 1: $18\sqrt{3}$
Option 2: $24\sqrt{3}$
Option 3: $36\sqrt{3}$
Option 4: $42\sqrt{3}$
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Correct Answer: $36\sqrt{3}$
Solution :
ABCDEF is a regular hexagon that is divided into 6 equilateral triangles with a side of 12 cm.
We know, Area of a regular hexagon $= 6 × \frac{\sqrt{3}}{4}$ × (side)
2
= $ 6 × \frac{\sqrt{3}}{4} ×12^2=216\sqrt3$ cm
2
Area of $\triangle$EOD = area of $\triangle$DOC = $\frac{1}{6}$ × (Area of hexagon) = $\frac{1}{6}×216\sqrt3=36\sqrt3$ cm
2
By symmetry, the area of $\triangle$EGD = the area of $\triangle$CGD = the area of $\triangle$CGO = the area of $\triangle$EGO
⇒ Area of $\triangle$ECD = 2 × area of $\triangle$CGD = the area of $\triangle$COD
= $36\sqrt3$ cm
2
Hence, the correct answer is $36\sqrt{3}$.
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