GATE ECE 2021 | GA | Question: 10

Question

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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

• A. $2:3$
• B. $3:4$
• C. $4:5$
• D. $5:6$

Answer 1

Given that, corners are cut from an equilateral triangle to produce a regular convex hexagon. The smaller triangles cut are equilateral triangles (If we cut off corners to create a regular hexagon, then each angle of the hexagon is $120^{\circ}$, and so each angle of every removed triangle is $60^{\circ}$ making these triangles equilateral)

Let the original triangle side be $3a.$

The area of regular convex hexagon $H_{1} = \frac{3\sqrt{3}}{2}s^{2},$ the area of equilateral triangle $ T_{1}= \frac{\sqrt{3}}{4}s^{2},$ where $s$ is the side length. 

Now, $H_{1} = \frac{3\sqrt{3}}{2}a^{2},A_{1} = \frac{\sqrt{3}}{4}(3a)^{2} = \frac{9\sqrt{3}}{4}a^{2}$

The required ratio $ = \dfrac{H_{1}}{A_{1}} = \dfrac{\frac{3\sqrt{3}}{2}a^{2}}{\frac{9\sqrt{3}}{4}a^{2}} = \frac{2}{3}.$

$\therefore$ The ratio of the area of the regular convex hexagon to the area of the original equilateral triangle is $2:3.$

So, the correct answer is $(A).$

Comments

We can also see that the pink triangles are also equilateral and of equal area therefore we can directly count the number of triangles in whole triangle and hexagon to find the ratio of area. 

$$\frac{\text{pink triangles}}{\text{total triangles}} = \frac{6}{9} $$ 

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We can consider the original triangle made up of 9 small triangle of equal area(let a). => area of original triangle=9a

Hexagon consists of 6 triangles. => area of hexagon = 6a.

Ratio would be 6a:9a or 2:3

Hence, option A .

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Yeah this was much easier and intuitive way because i don't get the formula of area of hexagon at that time...

Answer 2

i have very nice answer for this qeustion if there is regular hexagon then it's diagonals will create six equlateral traingles by joining them we can see the remaining three traingles are also the same as other six traingle so the ratio will be 2:3

Comments

Excellent 👌

 

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But how can we prove that triangles outside the hexagon are also equilateral ??

 

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You can see the angles are all 60, because a total line is 180 cut in 3 angles 3×60 so every angle in this figure is 60. This way you can prove it