Question

A square is of area \[200{\text{ }}sq.{\text{ }}m\]. A new square is formed in such a way that the length of its diagonal is \[\sqrt 2 \] times of the diagonal of the given square.Then the area of the new square formed is?

Answer 1

Hint:A square is a quadrilateral whose all four sides are equal and have right angles at the vertices. The square has two equal diagonals and the measure of the diagonal is obtained by the Pythagoras Theorem. Pythagoras Theorem is observed in a right triangle and the diagonal of the square divides the square in two right angles.

Complete step by step answer:
We have given that the area of the square is \[200{\text{ }}sq.{\text{ }}m\]. Let us consider the side of the square be\[a\]. Then we know that the area of a square is given as the square of its side.
Then we have \[{a^2} = 200\,sq.\,m\]
Taking positive square root both sides we get,
\[\sqrt {{a^2}} = \sqrt {200} \\
\Rightarrow a = 10\sqrt 2 \,m \\ \]
Now, the diagonal \[d\] of square of side \[a\] is given by \[d = a\sqrt 2 \]
So putting the value of \[a\] we get the diagonal of the square as
\[d = 10\sqrt 2 \times \sqrt 2 \\
\Rightarrow d = 20\,m \\ \]
According to question the length of the diagonal \[D\] of the new square is \[\sqrt 2 \] time the diagonal of the original square
Hence on multiplying by \[\sqrt 2 \] we get the diagonal of the new square as
\[D = 20 \times \sqrt 2 m \\
\Rightarrow D= 20\sqrt 2 \,m \\ \]
Let the side of the new square obtained be \[A\].So again using the relation between diagonal and the side of the square \[D = A\sqrt 2 \]. We get, side of the new square
\[A = \dfrac{{20\sqrt 2 }}{2}m \\
\Rightarrow A= 20m \\ \]
Therefore the area of the new square is given as the square of the side which is given as
\[\text{Area of the new square} = A \times A\]
Putting the value of\[A\], we get
\[\text{Area of the new square}= 20\,m \times 20\,m \\
\therefore \text{Area of the new square}= 400\,{m^2} \\ \]
Hence the area of the new square is \[400{\text{ }}sq.{\text{ }}m\].

Note:There are two diagonals of a square of equal lengths and they bisect each other but not at right angles. Rhombus also have the property that all the four sides of rhombus have equal length but the angle between two sides is not right angle and the diagonals of rhombus bisect each other at right angle. One should know this difference in order to differentiate between rhombus and a square. Values of the sides and multiple might vary in different types of questions.