Question

each side of the square below is 8 inches. what is the probability that a point chosen at random in the square is in the blue region? 0.25 0.33 0.66 0.75

Explanation:

Step1: Calculate area of square

The side length of the square is \( 8 \) inches (since \( 4 + 4 = 8 \)). The area of a square is given by \( A_{square}=s^2 \), where \( s \) is the side length. So, \( A_{square}=8^2 = 64 \) square inches.

Step2: Calculate area of blue region

The blue region consists of two triangles (or we can think of it as a combination that covers three - fourths of the square, but let's calculate it properly). The square can be divided into two rectangles of \( 4\times8 \) each. Looking at the blue triangles:
For the upper blue triangle: base can be considered as \( 8 \) and height as \( 4 \), area \( A_1=\frac{1}{2}\times8\times4 = 16 \).
For the lower blue triangle: base is \( 8 \) and height is \( 4 \), area \( A_2=\frac{1}{2}\times8\times4=16 \). Wait, no, actually, when we look at the figure, the total blue area: the square has side \( 8 \), the white triangle has a base of \( 8 \) and height of \( 4 \) (since the vertical segment is \( 4 \) and \( 4 \)). The area of the white triangle \( A_{white}=\frac{1}{2}\times8\times4 = 16 \). Then the blue area \( A_{blue}=A_{square}-A_{white}=64 - 16=48 \).

Step3: Calculate probability

Probability \( P=\frac{A_{blue}}{A_{square}}=\frac{48}{64}=\frac{3}{4} = 0.75 \).

Answer:

0.75