Question

the ratio of the lengths of the respective diagonals of two squares is 2:1.find the ratio of their areas

Answer 1

Let the coefficient be x.
Diagonals will be 2x and x.
Sides will be 2x/√2 and x/√2.
Now, Ratio of areas = (√2x)^2 /(x/√2)^2= 4:1.

Answer 2

Answer:

4:1

Step-by-step explanation:

Ratio of the lengths of the respective diagonals of two squares is 2:1.

Let the ratio be x

So, length of respective diagonals of two squares is 2x and x

Since Diagonal = $d=\sqrt{2} a$

Where a is the side of the square

So, In square 1

$2x=\sqrt{2} a$

$\frac{2x}{\sqrt{2}} = a$

Thus the side of the square of diagonal 2x is $\frac{2x}{\sqrt{2}}$.

So, In square 2

$x=\sqrt{2} a$

$\frac{x}{\sqrt{2}} = a$

Thus the side of the square of diagonal x is $\frac{x}{\sqrt{2}}$.

Area of square = $Side^2 = a^2$

So, ratio of their area = $\frac{(\frac{2x}{\sqrt{2}})^2}{(\frac{x}{\sqrt{2}})^2}$

                                 = $\frac{\frac{4x^2}{2}}{\frac{x^2}{2}}$

                                 = $\frac{4}{1}$

Thus the ratio of their area is 4:1