SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Get the angles of the hexagon
The internal angles of an hexagon is given as:
[tex]\begin{gathered} \frac{180(n-2)}{n} \\ n=6\text{ since hexagon has 6 sides} \\ So\text{ we have:} \\ \frac{180(6-2)}{6}=\frac{180(4)}{6}=\frac{720}{6}=120\degree \end{gathered}[/tex]
Therefore each angle of the hexagon is 120 degrees.
STEP 2: find the length of the sides
We remove the right triangles as seen below:
Using the special right triangles, we have:
STEP 3: find the area of the extracted triangle above
[tex]\begin{gathered} b=1,h=\sqrt{3} \\ Area=\frac{1}{2}\cdot1\cdot\sqrt{3}=\frac{\sqrt{3}}{2}units^2 \end{gathered}[/tex]
Since there are two right triangles, we multiply the area by 2 to have:
[tex]Area=2\cdot\frac{\sqrt{3}}{2}=\sqrt{3}[/tex]
There are two triangles(both sides), therefore the total area of the shaded area will be:
[tex]\sqrt{3}\cdot2=2\sqrt{3}[/tex]
STEP 4: Find the area of the whole hexagon
[tex]\begin{gathered} Area=\frac{3\sqrt{3}s^2}{2} \\ s=hypotenuse\text{ of the right triangle}=2 \\ Area=\frac{3\sqrt{3}\cdot4}{2}=6\sqrt{3} \end{gathered}[/tex]
STEP 5: Find the probability
[tex]\begin{gathered} Probability=\frac{possible\text{ area}}{Total\text{ area}} \\ \\ Possible\text{ area}=2\sqrt{3} \\ Total\text{ area}=6\sqrt{3} \\ \\ Probability=\frac{2\sqrt{3}}{6\sqrt{3}}=\frac{1}{3}=0.3333 \end{gathered}[/tex]
Hence, the probability that the dart hits one of the shaded areas is approximately 0.3333
