In a triangle ABC, with B being 90 degrees. BD= DC= 4. Find area of equilateral triangle DEF.D is mid-point of BC. Traingle DEF is inscribed in...

If triangle DEF is equilateral, therefore, the following
angles are congruent:DEF `-=` EDF`-=` DFE = 60
degrees.

Since the angle EDF measures 60 degrees, the
angles EDB nad FDC measure 60 degrees, also.

Since the
angle EDB measures 60 degrees and the point D represents the midpoint of BC, thererfore,
the segment ED is the midline of triangle ABC and according to midline theorem, it is
parallel to the base AC and it is half as long.

We'll
determine the length of AC.

sin 30 =
BC/AC

BC = BD + DC = 4+4 =
8

sin 30 = 1/2

1/2 =
8/AC

AC = 16 => ED = 16/2 =
8

Since the triangle DEF is equilateral, then all sides
have equal lengths.

Therefore, the area of DEF
is:

A = DE*EF*sin 60/2

A =
8*8*`sqrt(3)` /4

A = 16`sqrt(3)` square
units.

Therefore, the area of triangle DEF is
of 16`sqrt(3)` square units.