Question

10. CD and GH are respectively the bisectors
of ZACB and Z EGF such that D and H lie
on sides AB and FE of A ABC and A EFG
respectively. If A ABC - AFEG, show that:
CD AC
(1) GH
FG
(ii) ADCB - AHGE
(iii) ADCA - AHGF​

Answer 1

Given, CD and GH are respectively the bisectors of ∠ACB and ∠EGF such that D and H lie on sides AB and FE of ΔABC and ΔEFG respectively.  

(i) From the given condition,

ΔABC ~ ΔFEG.

∴ ∠A = ∠F, ∠B = ∠E, and ∠ACB = ∠FGE  

Since, ∠ACB = ∠FGE

∴ ∠ACD = ∠FGH (Angle bisector)

And, ∠DCB = ∠HGE (Angle bisector)  

In ΔACD and ΔFGH,

∠A = ∠F  

∠ACD = ∠FGH  

∴ ΔACD ~ ΔFGH (AA similarity criterion)

⇒CD/GH = AC/FG  

(ii) In ΔDCB and ΔHGE,  

∠DCB = ∠HGE (Already proved)

∠B = ∠E (Already proved)  

∴ ΔDCB ~ ΔHGE (AA similarity criterion)

(iii) In ΔDCA and ΔHGF,

∠ACD = ∠FGH (Already proved)

∠A = ∠F (Already proved)

∴ ΔDCA ~ ΔHGF (AA similarity criterion)