Maths-

General

Easy

Question

# The diagonal of a square is 4 √2 cm. Find the length of another diagonals if the diagonal of another square whose area is double that of the first square.

Hint:

### Area of a square = side2

All angles in a square are equal to 90°

Diagonal of a square divides it into 2 right angled triangles

## The correct answer is: ∴ Length of the diagonal of another square is 8 cm

### Step-by-step solution:-

In the adjacent diagram, we can see that the diagonal divides the given square into 2 right angled triangles.

Also, diagonal of the square becomes the hypotenuse of the 2 triangles.

Hypotenuse of the given triangles = diagonal of the given square = 4 √2 cm

Let the sides of the given square be x cm.

∴ side of the triangles (other than hypotenuse) = x cm.

By applying Pythagorean theorem, For a right angled triangle-

Hypotenuse2 = sum of the squares of the remaining 2 sides

∴ (4 √2)^{2} = x^{2} + x^{2}

∴ 16 × 2 = 2 x^{ 2}

∴ 32 = 2 x 2

i.e. 2 x^{ 2} = 32

∴ x^{2} = 32 / 2

∴ x^{2} = 16

∴ x = 4 .............................. (Taking square root both the sides) ........................ (Equation i)

Area of the given square = side^{2}

∴ Area of the given square = 4^{2} ...................................................................................... (From Equation i)

∴ Area of the given square = 16 cm^{2} ................................................................................... (Equation ii)

As per given information-

Area of another square = 2 × Area of original square

∴ Area of another square = 2 × 16 ..................................................................................... (From Equation ii)

∴ Area of another square = 32

∴ Side^{2} = 32 ........................................................................................... (Area of a square = side^{2})

∴ Side^{2} = 32

∴ Side = 4 √2 ......................................................................................... (Equation iii)

Apllying Pythagorean theorem, For a right angled triangle-

Hypotenuse^{2} = sum of the squares of the remaining 2 sides

∴ Hypotenuse^{2} = (4 √2)^{2} + (4 √2)^{2} ...................................................................................... (From Equation iii)

∴ Hypotenuse^{2} = 2 (4 √2)^{2}

∴ Hypotenuse^{2} = 2 (16 × 2)

∴ Hypotenuse^{2} = 2 (32)

∴ Hypotenuse^{2} = 64

∴ Hypotenuse = 8 ......................................................................................................... (Taking square root both the sides)

Final Answer:-

∴ Length of the diagonal of another square is 8 cm.

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