Question 11 of 14advanced🔧 ApplyLong Answer4 marks

CD and GH are respectively the bisectors of ∠ACB\angle ACB and ∠EGF\angle EGF such that D and H lie on sides AB and FE of ΔABC\Delta ABC and ΔEFG\Delta EFG respectively. If ΔABC∼ΔFEG\Delta ABC \sim \Delta FEG, show that:

(i) CDGH=ACFG\dfrac{CD}{GH} = \dfrac{AC}{FG}

(ii) ΔDCB∼ΔHGE\Delta DCB \sim \Delta HGE

(iii) ΔDCA∼ΔHGF\Delta DCA \sim \Delta HGF

(i)

CD/GH = AC/FG

Answer

CD/GH = AC/FG is proved

(ii)

Δ DCB ∼ Δ HGE

Answer

ΔDCB ∼ ΔHGE is proved

(iii)

Δ DCA ∼ Δ HGF

Answer

ΔDCA ∼ ΔHGF is proved

Explanation

This question tests the application of similarity criteria and angle bisector properties. The key insight is that when two triangles are similar, their corresponding angles are equal, and therefore the angles formed by their angle bisectors are also equal.

This allows us to establish AA similarity for the smaller triangles formed by the angle bisectors.

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