Question

In figure a d e f g is a square and angle BAC is equal to 90 degree show that the square is equal to BD into ec

Answer 1

In Δ AGF and Δ DBG,
∠ AGF = ∠ GBD (Corresponding angles)
∠ GAF = ∠ BDG = 90° each
So, Δ AGF ~ Δ DBG (Proved By AA similarity)


(2) In Δ AGF and Δ EFC,
∠ AFG = ∠ FCE (Corresponding angles)
∠ GAF = ∠ CEF = 90° each
So, Δ AGF ~ Δ EFC (Proved by AA similarity)
(3) In Δ DBG and Δ EFC,
∠ DBG = ∠ ECF = (Corresponding angles)
∠ BDG = ∠ CEF = 90° each
So, Δ DBG ~ Δ EFC (Proved by AA similarity)


(4) In Δ AGF and Δ DBG,
∠ AGF = ∠ GBD (Corresponding angles)
∠ GAF = ∠ BDG = 90° each
∴ Δ AGF ~ ΔDBG .....(1)
Similarly, Δ AFG ~ Δ ECF (AA similarity) ....(2)
From (1) and (2), we get
Δ DBG ~ Δ ECF
⇒ BD/EF = BG/FC = DG/EC
BD/EF = DG/EC
EF × DG = BD × EC ....(3)


Also DEFG is a square ⇒ DE = EF = FG = DG ....(4)


From (3) and (4), we get
DE² = BD × EC
Hence proved.