Question

in triangle abc, if bd is perpendicular to ac and bc^2= 2ac*cd, then prove that ab=ac

Answer 1

We have,
BC = 2 AC.CD
⇒ BD + CD = 2 AC.CD [Using pythagoras theorem in Δ BCD]
Again, in Δ ABD by pythagoras theorem, we have
AB = BD + AD
⇒ BD = AB - AD
So, AB - AD + CD = 2 AC.CD
⇒ AB - (AC - CD) + CD = 2 AC.CD
⇒ AB - AC - CD + 2 AC.CD + CD = 2 AC.CD
⇒ AB - AC = 0
⇒ AB = AC
⇒ AB = AC [Hence proved]