Mathematics
In a △ABC, ∠A = 90°, CA = AB and D is a point on AB produced. Prove that
DC2 - BD2 = 2AB × AD.
Pythagoras Theorem
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Answer
In right angle triangle ACD,
DC2 = CA2 + AD2 (Pythagoras theorem)
DC2 = CA2 + (AB + BD)2
DC2 = CA2 + AB2 + BD2 + 2AB.BD
DC2 - BD2 = AB2 + AB2 + 2AB.BD [∵ CA = AB]
DC2 - BD2 = 2AB2 + 2AB.BD
DC2 - BD2 = 2AB(AB + BD)
From figure, AB + BD = AD
DC2 - BD2 = 2AB.AD
Hence, proved that DC2 - BD2 = 2AB.AD.
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