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Question -

D and E are pointson the sides CA and CB respectively of a triangle ABC right angled at C. Provethat AE2 + BD2 = AB2 + DE2.



Answer -

Given, D and E are points on the sides CA andCB respectively of a triangle ABC right angled at C.

By Pythagoras theorem in ╬ФACE, we get

AC2 + CE2 =AE2 тАжтАжтАжтАжтАжтАжтАжтАжтАжтАжтАжтАжтАжтАжтАжтАж.(i)

In ╬ФBCD, by Pythagoras theorem, we get

BC2 + CD2 =BD2 тАжтАжтАжтАжтАжтАжтАжтАжтАжтАжтАжтАж..(ii)

From equations (i) and (ii),we get,

AC2 + CE2 +BC2 + CD2 = AE2 +BD2 тАжтАжтАжтАж..(iii)

In ╬ФCDE, by Pythagoras theorem, we get

DE2 = CD2 +CE2

In ╬ФABC, by Pythagoras theorem, we get

AB2 = AC2 +CB2

Putting the above two values in equation (iii), we get

DE2 + AB2 = AE2 + BD2.

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