Question

 

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. If ΔABC ~ ΔFEG, Show that

1. `("CD")/("GH") = ("AC")/("FG")`
2. ΔDCB ~ ΔHGE
3. ΔDCA ~ ΔHGF

 

Answer 1

Given:

(i) ΔABC ∼ ΔFEG 

∠A = ∠F             ...(1)

∠B = ∠E             ...(2)

∠C = ∠G            ...(3)

The sides opposite to equal angles are essentially equal.

`"CD"/"GH" = "AC"/"FG"`

(ii) In ΔDCB and ΔHGE

In equation (2), 

∠B = ∠E

In equation (3), 

∠C = ∠G 

∠BCD = ∠EGH

AA by similarity criterion

∆DCB ∼ ∆HGE

(iii) In ΔDCA and ΔHGF

In Equation (1)

∠A = ∠F

In equation (3), 

∠C = ∠G

∠ACD = ∠FGH

AA by similarity criterion

∆DCA ∼ ∆HGF