Q) In the given figure, CD is perpendicular bisector of AB. EF is perpendicular to CD. AE intersects CD at G. Prove that CF/CD = FG/DG.

Ans: 

Given that:

CD is perpendicular bisector of AB,

AD = BD, ∠ CDB = ∠ GDA = 90⁰

EF is perpendicular bisector of CD,

∠ EFC = ∠ EFG = 90⁰

Let’s look at Δ CEF and Δ CBD:

∠ EFC = ∠ BDC = 90⁰   (from given information)

∠ CEF = ∠ CBD            (corresponding angles)

Therefore, Δ CEF Δ CBD (by AA similarity identity)

Hence, ………. (i)

Since AD = BD, therefore,

Hence, …………. (ii)

Let’s look at Δ EFG and Δ GDA:

∠ EFG = ∠ GDA = 90⁰   (from given information)

∠ EGF  = ∠ AGD           (Vertically Opposite angles)

Therefore, Δ EFG Δ GDA (by AA similarity identity)

Hence, ….. (iii)

From equation (ii) and (iii), we get:

Hence Proved.

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