Here,

Side of the Larger square = 1 inch

Therefore Area Large square = \(1^2 =1\)

But we know Area of a square =\(\frac{diagonal^2}{2}\)

So 1 = \(\frac{diagonal^2}{2}\)

or diagonal = \(\sqrt{2}\)

And we know
smaller square region is \(\frac{2}{3}\) the area of the larger square region

So it can be written as \(\frac{(Diagonal smaller square)^2}{2}= \frac{2}{3} * 1 = \frac{2}{3}\) (Since the area of the Larger square = 1)

or \((diagonal of smaller square)^2 = \frac{4}{3}\)

or diagonal of smaller square = \(\frac{\sqrt4}{\sqrt3} = \frac{2}{(\sqrt3)} = \frac{2}{\sqrt3} * \frac{\sqrt3}{\sqrt3} = \frac{2\sqrt3}{3}\)

Now,
Diagonal of the larger square is longer than the diagonal of the smaller square = \(\sqrt2 - \frac{2\sqrt3}{3}\)