Question

in triangle abc,angle b is 90° and d is the mid point of bc. prove that bc^2 = 4(ad^2-ab^2)

Answer 1

Given: Triangle ABC, right-angled at B, D is the midpoint of BC

Prove: AC2 = 4 AD2 - 3 AB2

Proof:

In Triangle ABC,

AC2 = AB2 + BC2= AB2 + (BD+DC)2 = AB2 + (2BD)2 = AB2 + 4BD2------------------- 1

In Triangle ABD,

AD2 = AB2 + BD2 ------------------- 2

Subtract 2 from 1,

AC2 - AD2 = AB2 - AB2 + 4BD2 - BD2

AC2 - AD2 = 3BD2 = 3(AD2 - AB2) = 3AD2 - 3AB2

AC2 = 3AD2 - 3AB2 + AD2

AC2 = 4AD2 - 3AB2

Hence Proved

Answer 2

Given: In △ABC, ∠B = 90° and D is the mid-point of BC.

To Prove: AC2 = AD2 + 3CD2

Proof:

In △ABD,

AD2 = AB2 + BD2

AB2 = AD2 - BD2 .......(i)

In △ABC,

AC2 = AB2 + BC2

AB2 = AC2- BD2 ........(ii)

Equating (i) and (ii)

AD2 - BD2 = AC2 - BC2

AD2 - BD2 = AC2 - (BD + DC)2

AD2 - BD2 = AC2 - BD2- DC2- 2BDx DC

AD2 = AC2 - DC2 - 2DC2 (DC = BD)

AD2 = AC2 - 3DC2