In a triangle abc, ad is drawn perpendicular to bc. prove that ab2 - bd2 = ac2 - cd2.

Given: In triangle ABC , AD is drawn perpendicular to BC.

Since AD is drawn perpendicular to BC, it creates two right triangles: ADB and ADC.

Prove that:

Pythagoras triangle for right angle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

In a right angle triangle ADB;

                [By Pythagoras theorem]

or

                     .......[1]

Now, in right angle triangle ADC;

               [By Pythagoras theorem]

or we can write this as;

                   ......[2]

Substituting the equation [1] in [2] we get;

           hence proved!

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