Answer:
Area of trapezoid = 67.6 square units
Step-by-step explanation:
Area of a trapezoid is given by the expression,
Area of the trapezoid = [tex]\frac{1}{2}(b_1+b_2)h[/tex]
Here, [tex]b_1[/tex] and [tex]b_2[/tex] are the parallel sides of the given trapezoid.
And '[tex]h[/tex]' = Height between the parallel sides
From the given triangle ABE,
m(∠ABE) = m(∠ABC) - m(∠EBC)
m(∠ABE) = 120° - 90°
= 30°
By applying cosine rule in the given triangle,
cos(30°) = [tex]\frac{\text{Adjacent side}}{\text{Hypotenuse}}[/tex]
[tex]\frac{\sqrt{3} }{2}=\frac{BE}{AB}[/tex]
[tex]\frac{\sqrt{3} }{2}=\frac{BE}{6}[/tex]
BE = [tex]3\sqrt{3}[/tex] units
By applying sine rule in ΔABE,
sin(30°) = [tex]\frac{\text{Opposite side}}{\text{Hypotenuse}}[/tex]
[tex]\frac{1}{2}=\frac{AE}{AB}[/tex]
[tex]\frac{1}{2}=\frac{AE}{6}[/tex]
AE = 3 units
Length of [tex]b_1=BC=10[/tex]
Length of [tex]b_2=AD=(AE+EF+FD)[/tex] [AE = FD, since given trapezoid ABCD is an isosceles trapezoid]
[tex]b_2=3+10+3[/tex]
[tex]b_2=16[/tex]
Height between the parallel sides [tex]h=3\sqrt{3}[/tex]
Area of the trapezoid = [tex]\frac{1}{2}(BC+AD)BE[/tex]
= [tex]\frac{1}{2}(10+16)(3\sqrt{3})[/tex]
= [tex]39\sqrt{3}[/tex]
= 67.6 square units
