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 triangle ABC with angle A being 30 degrees.
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.