CAT 2017 Quantitative Ability

Question : 29 of 34

Marks: +1, -0

In the adjoining figure ABC is an equilateral triangle inscribing a square of maximum possible area. Again in this square there is an equilateral triangle whose side is same as that of the square. Further the smaller equilateral triangle inscribes a square of maximum possible area. What is the area of the innermost square if the each side of the outermost triangle be 0.01 m?

$(873-504√{3}) \;{cm}^{2}$
$(738-504√{3}) \; {cm}^{2}$
$(873-405√{2})\;{cm}^{2}$
None of these

Validate

Solution:

P C Q

is also an equilateral triangle

∴ \;\; P C=P Q=P M=a

∴ \;\; \;{a}/{P A}=\;{√{3}}/{2}

∴ \;\; P A=\;{2 a}/{√{3}}

∴ \;\; A C=A P+P C=\;{2 a}/{√{3}}+a=1 {cm}

⇒ \;\; a=\;{√{3}}/{(2+√{3})}=√{3}(2-√{3})

Now, in figure (iii)

P M=M T=a

Let the each side of square

R S Y X

K,

then

R T=K

also (since RTS is an equilateral triangle)

∴ \;\; \;{K}/{R M}=\;{√{3}}/{2}

∴ \;\; RM=\;{2 K}/{√{3}}

∴ \;\; MT=R⊤+R M=K+\;{2 K}/{√{3}}

MT=\;{(√{3}+2)}/{√{3}} K

but

M T=a

∴ \;\; a=(\;{√{3}+2}/{√{3}}) K

∴ \;\; K=\;{√{3} a}/{(√{3}+2)}

But

a =√{3}(2-√{3})

∴ \;\; K=\;{√{3}}/{(√{3}+2)}

\;\; =\;{3(2-√{3})}/{(2+√{3})} × \;{(2-√{3})}/{(2-√{3})}

⇒ \;\; K=\;{3(2-√{3})^{2}}/{1}=3(7-4 √{3})

∴ Area of square,

R S Y X \;\; =K^{2}=[3(7-4 √{3})]^{2}

K^{2} \;\; =9(49+48-56 √{3})

K^{2} \;\; =(873-504 √{3}) {cm}^{2}

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