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Maths-

General

Easy

Question

D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C . Prove that
$A E squared plus B D squared equals A B squared plus D E squared$

The correct answer is: Hence proved

Solution :-
Aim :- Prove that $A E squared plus B D squared equals A B squared plus D E squared$

Hint :- Applying pythagoras theorem to both triangles ABC,DEC,DBC and AEC. find the equation, add them and substitute proper conditions to prove the condition
Explanation(proof ) :-
Applying pythagoras theorem to triangle ABC , we get
$A B squared equals A C squared plus B C squared minus bold Eql$
Applying pythagoras theorem to triangle DEC , we get
$D E squared equals D C squared plus E C squared$ — Eq2
Applying pythagoras theorem to triangle DBC , we get
$D B squared equals B C squared plus D C squared minus bold Eq 3$
Applying pythagoras theorem to triangle AEC , we get
$A E squared equals A C squared plus E C squared minus bold Eq 4$

Adding Eq 1 and 2,we get
$A B squared plus D E squared equals D C squared plus E C squared plus A C squared plus B C squared minus bold Eq 5$
Adding 3 and 4 we get ,
$D B squared plus A E squared equals A C squared plus E C squared plus B C squared plus D C squared minus bold Eq bold 6$
But substitute Eq6 in Eq5
We get $A E squared plus B D squared equals A B squared plus D E squared$
Hence proved

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