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$NM=6 \Rightarrow MF^2=\sqrt{10^2-6^2}=8$

$EA=MN-HB=3 \therefore \frac{MN}{EA}=\frac{MF}{EF}=2 \Rightarrow EF=4$

Shaded area is $ABCD \text{ area}-\triangle AFD\text{ area}$ so

$ABCD$ area is $(NH-ME)(2(HG-HB))=16(6)=96$

$\triangle AFD$ area is $\frac{2(AE)(EF)}{2}= \frac{2(3)(4)}{2}=12$

$\therefore$ shaded area is $96-12=\boxed{84}$

Denote the rectangle ABCD clockwise (from top-left). Extend the strip until it reaches AD. Now the area of this extended strip = 6 (width) $\times$ 20 = 120 (#1). Now we draw an altitude on the triangle and use Pythagoras's theorem to show that it's altitude is 8 (using the 2{3,4,5} Pythagorean triple) $\Rightarrow$ area of triangle = $\frac{12\times8}{2}$ = 48 (#2). using (#1) we subtract this area from our extended strip: 120-48 = 72. However, we counted the two outer triangles '-1' times, so we slide them together to form a triangle that is half the scale factor of the large triangle, so it has a quarter of the area. Adding this on gives: Area of shaded strip = 72 + $\frac{48}{4}$ = 72+12 = $\boxed{84}$

close the shaded area as rectangle.

Area: Area of rectangle (ABCD) - Area of Triangle (ADE)

i.e (20 * 16) - (6 * 4)/2 = 96 -12 = 84

You can view the shaded area as two trapezoids.

We know the area of a trapezoid is $\frac { a+b }{ 2 } h$ . To find base a and b, you need the height of the triangle left of the shaded region.

Split that triangle into two right triangles and you have a triangle with base 6 (12 / 2) and hypotenuse 10. Solve with the Pythagorean Theorem and you get the height of the triangle is 8.

Subtract that length from the total length of the rectangle (20) and you get the shorter base length is equal to 12. The other length extends to half the height of the triangle so it is length 16. The height of the trapezoid is 3 because it is 12 - 6 to get the height of the shaded area, divided by two to get the height of a trapezoid.

Solve for the trapezoid's area: $\frac { 16+12 }{ 2 } \times 3\quad =\quad 42$

Since there are two triangles in the shaded area, the shaded area is $42\times 2$

There it is: 84