Explanation:
To understand the ratio of the area of a square to that of the square drawn on its diagonal, let's consider a square with side length 's'.

Area of the square:
The area of a square is given by the formula A = s^2, where 's' is the length of each side of the square.

Diagonal of the square:
In a square, the diagonal divides the square into two congruent right triangles. Using the Pythagorean theorem, we can find the length of the diagonal.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (diagonal in this case) is equal to the sum of the squares of the other two sides (sides of the square).

In a square, the two sides are of equal length 's', so the Pythagorean theorem can be written as:
s^2 + s^2 = d^2
2s^2 = d^2
Taking the square root of both sides, we get:
d = sqrt(2)*s

Area of the square drawn on the diagonal:
The square drawn on the diagonal has side length equal to the length of the diagonal, which is sqrt(2)*s.
So, the area of the square drawn on the diagonal is given by:
A' = (sqrt(2)*s)^2
A' = 2s^2

Ratio of the areas:
The ratio of the area of the square to that of the square drawn on its diagonal can be calculated as:
A/A' = (s^2)/(2s^2)
A/A' = 1/2

Therefore, the ratio of the area of a square to that of the square drawn on its diagonal is 1:2, which corresponds to option 'B'.

Answer:
The correct answer is option 'B' - 1:2.