Question

abc is a triangle in which ab=ac and d is point on bc prove that ab²-ad²=bd×cd

Answer 1

In a triangle ABC, AB=AC. D is any point on BC. Show that AB 2 -AD 2 = BD . CD

Solution: 

To prove: AB 2 -AD 2 = BD . CD

Construction: Draw AE perpendicular to BC.

Proof: In tr. AEB and tr. AEC, 

AB= AC [given]

∠AEB = ∠AEC = 90°

AE= AE [common]

∴ΔAEB  similar to ΔAEC [By RHS]

⇒BE= EC[cpct]

in tr. ABE,

 ∠E= 90°
AB^2=AE^2+BE ^2..........1. by Pythagoras
E= 90
AD^2. = AE^2 +DE^2. ..2. by Pythagoras

on subtracting (ii) from (i). we get,
AB^2-AB^2 = AE^2+ BE^2 - AE^2 + DE^2
= BE^2. + DE^2
=( EC + ED) BD
= CD. BD

Answer 2

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