Question

In a triangle BD is perpendicular to AC . Prove that AB²-BC²+AC²= 2AC.AD​

Answer 1

GIVEN: In triangle ABC :

angle B =90°; BD perpendicular AC

TO PROVE: AB²- BC²+ AC² = 2*AC*AD

PROOF: In triangle ABC and triangle ADB

Angle A = Angle A (common)

Angle B = Angle D (90°)( BD perpendicular AC)

=> Triangle ABC ~ Triangle ADB (A.A)

=> AB = BC = AC (EQ...1)

AD DB AB

Similarly, Triangle ABC ~ Triangle BDC

=> AB = BC = AC (EQ...2)

BD DC BC

From eq. 1 we have AB = AC

AD AB

=> AB² = AC* AD

Similarly from eq. 2 we have

BC = AC

DC BC

=> BC²= AC* DC

WE KNOW THAT AC² = AC*AC

= AC( AD + DC) ( FROM FIGURE)

THUS;

AB²-BC² + AC² =(AC*AD) - (AC*DC ) +( AC(AD +DC)

= AC*AD - AC*DC + AC*AD + AC*DC

= AC*AD + AC*AD

= 2*AC*AD

Hope it helps....