Question

In the given figure DEllBC and CDllEF prove that AD2=ABXAF
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Answer 1

Answer:

We have to prove that AD2=AF×AB

Consider △ABC,DE∥BC by basic proportionality theorem. 

If a line is parallel to a side of a triangle which intersects the other sides into two distinct points, then the line divides the sides in proportion. 

i.e, DBAD=ECAC (i)

From △ADC,EF||DC

∴, By Basic Proportionality Theorem, FDAF=ECAE.(ii)

From (i) and (ii), DBAD=FDAF

⇒ADDB=AFFD[Taking Reciprocal](iii)  

Adding (i) and (iii), 

ADDB+AD=AFFD+AF

ADAB=AFAD

∴(AD)2=AF×AB