Question

If the length of the diagonal of a square is \[10\sqrt 2 \] cm, then its perimeter is
A.10 cm
B.\[40\sqrt 2 \]cm
C.20 cm
D.40 cm

Answer 1

Hint: Here, we need to find the perimeter of a square whose diagonal length is given. We will use the Pythagoras theorem in this case to find the length of the side of the square taking diagonal as the hypotenuse. Then we will find the perimeter of the square using the length of its side.

Formula used:
We will use the following formulas:
Pythagoras theorem: \[{H^2} = {P^2} + {B^2}\], where \[H\] is the hypotenuse, \[P\] is the perpendicular and \[B\] is the base.
Perimeter of a square = \[4 \times \]length of the side of the square

Complete step-by-step answer:
We have been given the length of the diagonal of a square. We will find the length of its side as follows:
Let us assume that the side of the square ABCD is \[x\] cm.

Since each internal angle of a square is \[90^\circ \], we shall apply the Pythagoras theorem to the right-angled triangle BCD. The Pythagoras theorem states that the sum of the squares on the two sides of a right-angled triangle is equal to the square on the hypotenuse. In \[\Delta BCD\], BD is the hypotenuse, BC is the perpendicular and CD is the base.
Applying the Pythagoras theorem to \[\Delta BCD\], we have
\[B{C^2} + C{D^2} = B{D^2}\]
Substituting \[BC = CD = x\] cm and BD \[ = \] \[BD = 10\sqrt 2 \] cm in the above equation, we get
\[ \Rightarrow {x^2} + {x^2} = {\left( {10\sqrt 2 } \right)^2}\]
Adding the like terms on the LHS and applying exponents on the RHS, we get
\[ \Rightarrow 2{x^2} = {\left( {10} \right)^2}{\left( {\sqrt 2 } \right)^2}\]
\[ \Rightarrow 2{x^2} = 100 \times 2\]
 Multiplying the terms, we get
\[ \Rightarrow 2{x^2} = 200\]
Dividing both sides by 2, we get
\[\begin{array}{l} \Rightarrow \dfrac{{2{x^2}}}{2} = \dfrac{{200}}{2}\\ \Rightarrow {x^2} = 100\end{array}\]
Taking square root, we get
 \[ \Rightarrow x = 10\] cm
Hence, the length of the side of a square is 10 cm.
Now, we shall find the perimeter of the square using the formula
 Perimeter of a square \[ = 4 \times \] length of the side of the square
 \[ \Rightarrow \] Perimeter of a square \[ = 4 \times 10 = 40\] cm
$\therefore $ The perimeter of the square is 40 cm and the correct option is D.

Note: We know that all the sides of the square are equal. Therefore, the perimeter is equal to 4 times the length of the sides. We can also find the side of the square as follows:
If the length of the side of a square is \[x\] cm, then the length of its diagonal is \[x\sqrt 2 \] cm.
Here, the length of diagonal \[ = 10\sqrt 2 \] cm
$\therefore $ The length of side \[ = \dfrac{{10\sqrt 2 }}{{\sqrt 2 }} = 10\] cm