GATE Mechanical 2021 Set 2 | GA | Question: 9

Question

Consider a square sheet of side $1$ unit. The sheet is first folded along the main diagonal. This is followed by a fold along its line of symmetry. The resulting folded shape is again folded along its line of symmetry. The area of each face of the final folded shape, in square units, equal to _________

• A. $\frac{1}{4}$

• A. $\frac{1}{8}$

100. $\frac{1}{16}$

500. $\frac{1}{32}$

Comments

video solution

https://youtu.be/omY0NKJIgks?si=Uu-g-6yPwSb-7p-C&t=1619

Answer 1

Given that a square sheet has a side length of $1$ unit.

Now, the sheet is first folded along the main diagonal. 

Again, this is followed by a fold along its line of symmetry. And this operation is performed two times.

We get the final folded sheet.

Let the height of the triangle be $h$ units.

Using the Pythagorean theorem, we get.

$\left(\frac{1}{\sqrt{2}}\right)^{2} = h^{2} + \left(\frac{1}{2}\right)^{2}$

$\implies \frac{1}{2} = h^{2} + \frac{1}{4}$

$\implies h^{2} = \frac{1}{2} – \frac{1}{4}$

$\implies h^{2} = \frac{2 – 1 }{4} = \frac{1}{4}$

$\implies h = \frac{1}{2}$

Now, the area of triangle $\Delta = \frac{1}{2} \times \text{Base} \times \text{Height}$

$\implies \Delta = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}\;\text{unit}^{2}.$

$\textbf{Short Method:}$ Area follows the symmetry.

• Initially the area of square $ = 1$
• Folded first time, then the area of square $ = \frac{1}{2}$
• Folded second time, then the area of square $ = \frac{1}{4}$
• Folded third time, then the area of square $ = \frac{1}{8}$

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

Comments

Question :-

The area of each face of the final folded shape, in square units, equal to-

I got it how we are getting 1/8 square units but the question is asking about each face. when we have folded the last time then their have to be two faces - front or back side. Just like cube has 6 faces.

So, isn’t area has to be 2* 1/8 = ¼ square units.??

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@ASNR1010 question is asking about area of EACH face . which means one face . 

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Your solution is mathematically correct but I will prefer to go with intuation that the area is getting divided 1/2 times in every fold as in exam this much math for a amplitude question I feeling like it's not a good idea

Answer 2

Ans- B ) 1/8

Comments

same thought

Answer 3

Given that a square sheet has a side length of $1$ unit.

Now, the sheet is first folded along the main diagonal. 

Again, this is followed by a fold along its line of symmetry. And this operation is performed two times.

We get the final folded sheet.

Let the height of the triangle be $h$ units.

Using the Pythagorean theorem, we get.

$\left(\frac{1}{\sqrt{2}}\right)^{2} = h^{2} + \left(\frac{1}{2}\right)^{2}$

$\implies \frac{1}{2} = h^{2} + \frac{1}{4}$

$\implies h^{2} = \frac{1}{2} – \frac{1}{4}$

$\implies h^{2} = \frac{2 – 1 }{4} = \frac{1}{4}$

$\implies h = \frac{1}{2}$

Now, the area of triangle $\Delta = \frac{1}{2} \times \text{Base} \times \text{Height}$

$\implies \Delta = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}\;\text{unit}^{2}.$

$\textbf{Short Method:}$ Area follows the symmetry.

• Initially the area of square $ = 1$
• Folded first time, then the area of square $ = \frac{1}{2}$
• Folded second time, then the area of square $ = \frac{1}{4}$
• Folded third time, then the area of square $ = \frac{1}{8}$

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

Answer 4

https://gateoverflow.in/359492/gate-mechanical-2021-set-2-ga-question-9?show=414297#a414297
 

solution