Question

Diagonal of a square is \[5\sqrt 2 \] . Length of the side of the square is
\[\left( A \right)\] \[10\]
\[\left( B \right)\] \[5\]
\[\left( C \right)\] \[3\sqrt 2 \]
\[\left( D \right)\] \[2\sqrt 2 \]

Answer 1

Hint: We have to find the value of the side of the square when the diagonal of the square is given as \[5\sqrt 2 \]. We solve this question using the concept of the properties of squares and the Pythagoras theorem. First we will draw a square, and consider the length of the sides of the square to be \[x\]. Then, using the Pythagoras theorem we will find the value of the side of the square.

Complete step by step answer:
Given, The value of the diagonal of a square is \[5\sqrt 2 \] .
Let us consider the side of the square to be \[x\].
The construction of the required square is as follows:

Now, we have to find the value of \[x\] .
Let us consider the \[\Delta ABD\] .
Then the formula for the Pythagoras theorem is given as :
\[A{D^2} + A{B^2} = B{D^2}\]
Now, using the formula of Pythagoras theorem and substituting the values we can write the expression as
\[{x^2} + {x^2} = {\left( {5\sqrt 2 } \right)^2}\]
\[2{x^2} = 25 \times 2\]
On further simplifying , we can write the expression as :
\[{x^2} = 25\]
Taking square root , we get the value of \[x\] as :
\[x = \pm 5\]
So , we get the value of the side of the square as :
\[x = 5\]
Hence , we get the value of the side of the square with the value of diagonal \[5\sqrt 2 \] as \[5\] .
Thus, the correct option is \[\left( 2 \right)\].

Note:
We could also find the value of the side of the square from the \[\Delta BCD\] by using the same process of applying the Pythagoras theorem as done above.
Each diagonal divides the square into two congruent isosceles right triangles. The triangles such formed have a half of the area of a square , its legs are the sides of the square and hypotenuse equals to the length of the diagonal of a square.
We can also determine the area of the square using the diagonal of the squares. The formula that can be used is, \[Area{\text{ }}of{\text{ }}the{\text{ }}square = \dfrac{1}{2} \times {d^2}\], where \[d\] is the length of diagonal of the square.