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Corners are cut from an equilateral triangle to produce a regular convex hexagon as shown in the figure above. The ratio of the area of the regular convex hexagon to the area of the original equilateral triangle is

Corners are cut from an equilateral triangle to produce a regular convex hexagon as shown in the figure above. The ratio of the area of the regular convex hexagon to the area of the original equilateral triangle is Correct Answer 2 : 3

Formula used:

The area of the equilateral triangle = (√3/4) × a2

The area of the regular hexagon = (3√3/2) × b2

Here a → side of the equilateral triangle and b → Side of the regular hexagon

Concept used:

When the corner is cut from the equilateral triangle to produce the regular hexagon then the side of the triangle is divided into 3 equal parts and one part is the side of the hexagon.

Then, b = a/3

Calculation:

The area of the equilateral triangle = (√3/4) × a2

The area of the regular hexagon = (3√3/2) × b2

⇒ (3√3/2) × (a/3)2

⇒ (√3/6) × (a)2

The ratio of the regular hexagon to the equilateral triangle = {(√3/6) × (a)2}/{(√3/4) × a2}

⇒ 4/6 = 2 ∶ 3

∴ The required ratio is 2 ∶ 3

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