Question

20) In A ABC, AB = AC and BD I AC, prove that
BD 2 + CD 2 = 2AC​

Answer 1

Answer:

ABC is a isosceles triangle in which AB=AC

BD Perpendicular to AC

In △ABD

AB 2 =AD 2 +BD 2 ...(Pythagoras theoram)

∴AC 2 =AD 2 +BD 2 ....(as AB=AC) ...(1)

From diagram, we see that

⇒AC=CD+AD

⇒AC 2 =(CD+AD) 2

From (1) we get,

⇒(CD+AD) 2 =AD 2 +BD 2

CD 2 +AD 2 +2(CD×AD)=AD 2 +BD 2

CD 2 + 2(CD×AD)=BD 2 ...(AD square gets cancelled from both sides)

BD 2 −CD 2

=2(CD×AD)

Hence proved.