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