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.

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 = x² + x²
∴  16 × 2 =  2 x ²
∴  32 = 2 x 2
 i.e. 2 x ² = 32
∴  x² = 32 / 2
∴  x² = 16
∴  x = 4 .............................. (Taking square root both the sides) ........................ (Equation i)
Area of the given square = side²
∴ Area of the given square = 4² ...................................................................................... (From Equation i)
∴ Area of the given square = 16 cm² ................................................................................... (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
∴ Side² = 32 ........................................................................................... (Area of a square = side²)
∴ Side² = 32
∴ Side = 4 √2 ......................................................................................... (Equation iii)
Apllying Pythagorean theorem, For a right angled triangle-
Hypotenuse² = sum of the squares of the remaining 2 sides
∴ Hypotenuse² = (4 √2)² + (4 √2)² ...................................................................................... (From Equation iii)
∴ Hypotenuse² = 2 (4 √2)²
∴ Hypotenuse² = 2 (16 × 2)
∴ Hypotenuse² = 2 (32)
∴ Hypotenuse² = 64
∴ Hypotenuse = 8 ......................................................................................................... (Taking square root both the sides)
Final Answer:-
∴ Length of the diagonal of another square is 8 cm.