Question

In triangle ABC ,EF||AB and ar(CEF)=ar(EFBA).then find the ratio of CF and FB.

Answer 1

see diagram.

Draw a perpendicular from C onto AB to meet AB at G and EF at H.

Area ΔABC = Δ = 1/2 * AB * CG 
  = ΔEFBA + Δ CEF 
  = 2 * Δ CEF = EF * CH
=>  AB * CG =  2 EF * CH   --- (1)

ΔABC and Δ CEF are similar triangles.  So the ratios of their corresponding sides and altitudes are equal.

   EF/AB = CE / CA = CF / CB = CH / CG
  => CH = EF * CG / AB

 So (1) => AB * CG = 2 EF² * CG / AB
=> AB² = 2 EF²
=> AB = √2 EF
The ratio of the sides in the two triangles is √2.

 CF  = CB / √2
 FB = CB - CF = CB* (√2-1)/√2

CF / FB = 1/(√2-1)  = √2 + 1    by rationalizing the denominator.