In equilateral triangle ABC, points D, E, F are on AB, BC, CA, respectively, with AD = BE = CF = 1 and DB = EC = FA = sqrt(3). Find the area of triangle DEF.
+128475
A
1
D sqrt (3)
sqrt (3) F
1
B 1 E sqrt (3) C
DE = EF = DF.....so triangle DEF is also equilateral
And we can find DE using the Law of Cosines
DE^2 = (sqrt (3))^2 + 1^2 - 2 (sqrt(3) * 1 * cos (60°)
DE^2 = 3 + 1 - 2 sqrt (3)( 1/2)
DE^2 = 4 - sqrt(3)
So....the area of triangle DEF = (1/2) (DE)^2 * sin (60°) = (1/2) ( 4 - sqrt (3) ) ( √3/2) =
( √3 / 4) (4 - √3) =
[ √[3 - 3/4 ] units^2
