Let's see what might happen if we add an auxiliary line to our diagram as follows:

This tells us that: area of shaded region = (area of ∆ABC) - (area of ∆ABD)


Let's start by finding the length of AB.
To do this we can focus on the red triangle below.
Since we have a right triangle, we can apply the Pythagorean theorem to write: 4² + 4² = (side AB)²
Simplify: 32 = (side AB)²
Find the square root of both sides: √32 = side AB
Simplify: 4√2 = side AB
When we add this to our diagram we get:



At this point, we can see that ∆ABC is an equilateral triangle

Area of equilateral triangle \(= (\frac{\sqrt{3}}{4})(side^2)\)

So, the area of ∆ABC \(= (\frac{\sqrt{3}}{4})(4\sqrt{2})^2= (\frac{\sqrt{3}}{4})(32)\) \(=8\sqrt{3}\)




Area of triangle \(= \frac{(base)(height)}{2}\)
So, the area of ∆ABD \(= \frac{(4)(4)}{2}\) \(=8\)


We're now ready to answer the question...
area of shaded region = (area of ∆ABC) - (area of ∆ABD)

= \(8\sqrt{3}\) - \(8\)

= \(8(\sqrt{3}-1)\)

Answer: D

Cheers,
Brent