Question

The image below shows a square inscribed inside a square. The vertices of the gray shaded square bisect the 4 sides of the larger square

Answer 1

To find the probabilities, the first step is to find the areas of the squares

First find the area of the larger square

The length is 2x

A = (2x)^2 = 4x^2

Then we need to find the area of the shaded square

That will be a 45 45 90 triangle so we can find the length of the hypotenuse,h. The hypotenuse is sqrt(2) times the length of the side of the triangle

h = x sqrt(2)

The side length for the shaded square is x sqrt(2) so

A = (x sqrt(2))^2 = 2 x^2

P( shaded area) = shaded area/ total area

= 2 x^2 / 4 x^2 = 1/2