Mathematics
In triangle ABC, AB = AC and BD is perpendicular to AC. Prove that :
BD2 - CD2 = 2CD × AD
Pythagoras Theorem
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Answer
By formula,
By pythagoras theorem,
⇒ (Hypotenuse)2 = (Perpendicular)2 + Base2
In right-angled triangle ABD,
By pythagoras theorem,
⇒ AB2 = AD2 + BD2
⇒ AD2 = AB2 - BD2 …….(1)
From figure,
⇒ AC = AD + DC
Squaring both sides, we get :
⇒ AC2 = (AD + DC)2
⇒ AC2 = AD2 + DC2 + 2AD.DC
⇒ AC2 = AB2 - BD2 + DC2 + 2AD.DC [From equation (1)]
Substituting AB = AC, in above equation :
⇒ AC2 = AC2 - BD2 + DC2 + 2AD.DC
⇒ AC2 - AC2 + BD2 - DC2 = 2AD.DC
⇒ BD2 - DC2 = 2CD × AD.
Hence, proved that BD2 - DC2 = 2CD × AD.
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