Question

In a triangle ABC, AD is perpendicular to BC. Prove that :
AB² + CD² = AC² + BD².

Answer 1

When AD is drawn perpendicular to BC, twotriangles ADB and ADCare obtained.

Considering triangle ADB,

AB2= AD2+ BD2......(1) (using, Pythagoras theorem)

Considering triangle ADC,

AC2= AD2+ CD2(2) (using Pythagoras theorem)

Subtracting equation (2) from equation (1),

AB2-AC2= AD2+ BD2- (AD2+ CD2)

AB2- AC2= AD2+ BD2- AD2- CD2

AB2- AC2= BD2- CD2

AB² + CD² = AC² + BD² hence proved