Question

In triangle ABC,angle B= 90degree and D is the mid point of BC.Prove that AC^=AD^+3CD^

Answer 1

Sol:  D is the midpoint of BC. ⇒ AD = CD. Angle B is a right angled triangle. Consider ΔABC AC2= AB2 + BC2 [Pythagoras theorem] ⇒ AC2= AB2 + (2BD)2 ⇒ AC2 = AB2 + 4BD2 ----------- (1) Consider ΔABC AD2 = AB2 + BD2[Pythagoras theorem] ----------- (2)Subtracting equation (2) from (1), we get⇒ AC2 - AD2 = 3BD2 ⇒ AC2 - AD2 = 3CD2 [ Since BD = CD] ⇒ AC2 = AD2 + 3CD2Hence proved.