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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

 

 Dec 4, 2020
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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)

 

 

cool cool cool

 Dec 4, 2020
edited by CPhill  Dec 4, 2020

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