**ABC is a triangle in which ABAC = 90° and DEFG is a square, prove that DE2 BD x EC.**

###### Answers:

Given: ABC is a triangle in which ∠BAC = 90° and DEFG is a square.

To prove: DE2 = BD x EC.

Proof: In △AGF and △DBG,

∠AGF = ∠GBD (corresponding angles)

∠GAF = ∠BDG (each = 90‘)

∴△AGF ~ △DBG …..(i)

Similarly, △AFG ~ △ECF (AA Similarity)…..(ii)

From (i) and (ii), △DBG ~ △ECF.

BD/EF – BG/FC – DG/EC

BD/EF – DG/EC

EF × DG = BD × EC……(iii)

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

From (iii) and (iv), DE2 = BD × EC.