Question

If the diagonal of a square is ‘a’ units. What is the diagonal of the square, whose area is double that of the first square?

Answer 1

Hint: Use the property that the area of the square is equal to half of the square of the length of the diagonal of the square. It is also important to express the length of the new square in terms of ‘a’ before reporting the answer.

Complete step-by-step answer:
Let us start the solution by letting the length of the diagonal of the new square to be ‘x’ units. These can be diagrammatically represented as:

Now we know that the area of the square is equal to half of the square of the length of the diagonal of the square.
$\therefore \text{ Area of the old square = }\dfrac{{{a}^{2}}}{2}$
$\therefore \text{ Area of the new square = }\dfrac{{{x}^{2}}}{2}$
Now it is given in the question that the area of the new square is twice the old one. Therefore, this can be mathematically written as:
$\text{ Area of the old square = }\dfrac{\text{Area of the new square}}{2}$
$\Rightarrow \dfrac{{{a}^{2}}}{2}\text{ = }\dfrac{{{x}^{2}}}{2\times 2}$
$\Rightarrow {{x}^{2}}=2{{a}^{2}}$
Now if we take the square root of both the sides of the equation, we get
$x=\sqrt{2}a$
Therefore, the length of the diagonal of the square whose area is double the area of the first square is equal to $\sqrt{2}a$ units.

Note: We could have also solved the above question using the property that the diagonal of a square is $\sqrt{2}$ times of the length of its side followed by the use of the formula that the area of the square is equal to the square of the length of its side.