Question

In the figure above, midpoints of the sides of a square are joined to form another smaller square, and the same process is repeated 5 times. If the area of the smallest shaded square is 2, what is the area of the largest square?

Answer 1

Question : In the figure ( given in the question ) , midpoints of the sides of a square are joined to form another smaller square, and the same process is repeated 5 times. If the area of the smallest shaded square is 2, what is the area of the largest square ?

Solution :

Observe the diagram carefully ,

Let the side of the largest square be ' a '  units .

So , Area of the largest square = a² unit²

Side of the 2nd square is a/√2 units ( Use Pythagoras theorem ) or

Diagonal of 2nd square = side of 1st (largest) square ( Observe carefully )

∵ Area of the square = $\sf \dfrac{(Diagonal)^2}{2}$

➠ Area of 2nd square = a²/2 unit²

Likewise ,

Area of 3rd square = a²/4 unit²

Area of 4th square = a²/8 unit²

Area of 5th square = a²/16 unit²

Area of 6th square = a²/32 unit²

We are given that , Area of the smallest (6th) shaded square = 2 unit²

$:\implies \sf \dfrac{a^2}{32}=2$

$:\implies \sf a^2=64$

$:\implies \sf a^2=(8)^2$

$:\implies \sf a=8\ units$

So , Area of the largest square = a² = 8² = 64 unit²

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Answer 2

Step-by-step explanation:

Question : In the figure ( given in the question ) , midpoints of the sides of a square are joined to form another smaller square, and the same process is repeated 5 times. If the area of the smallest shaded square is 2, what is the area of the largest square ?

Solution :

Observe the diagram carefully ,

Let the side of the largest square be ' a ' units .

So , Area of the largest square = a² unit²

Side of the 2nd square is a/√2 units ( Use Pythagoras theorem ) or

Diagonal of 2nd square = side of 1st (largest) square ( Observe carefully )

∵ Area of the square = \sf \dfrac{(Diagonal)^2}{2}

2

(Diagonal)

2

➠ Area of 2nd square = a²/2 unit²

Likewise ,

Area of 3rd square = a²/4 unit²

Area of 4th square = a²/8 unit²

Area of 5th square = a²/16 unit²

Area of 6th square = a²/32 unit²

We are given that , Area of the smallest (6th) shaded square = 2 unit²

:⟹a=8 units

So , Area of the largest square = a² = 8² = 64 unit²

★ ═════════════════════ ★