The figure below shows a triangle ABC which area is 72cm2. If AD: DB = BE: EC =CF: FA =1: 5, find the area of triangle DEF
+128474 3 * area of ABC is given by
(1/2) [ AC * BC sin C ] + (1/2) [ AB * BC sin B ] + (1/2) [ AB * AC sin A] = 216
Area of triangle EFC = (1/2) [ (1/6)AC * (5/6)BC sin C ] = (1/2)(1/6)(5/6) [ AC * BC sin C ] =
( 5/36) [ (1/2) [ AC * BC sin C ]
Area of triangle DBE = (1/2) [ (5/6)AB * (1/6) BC sin B ] = (5/36) [ (1/2) AB *BC sin B]
And we can intuit that the area of triangle DAF = (5/36) [ (1/2) AB * AC sin A ]
Therefore
(5/36) ( [ (1/2) [ AC * BC sin C ] + [ (1/2) AB *BC sin B] + [ (1/2) AB * AC sin A ] ) =
(5/36) [ 216 ] = 5 ( 216/36) = 5 * 6 = 30 = area of 3 small triangles combined
So [ DEF ] = 72 - 30 = 42 (cm^2)
