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Maths-

General

Easy

Question

$B D text is an altitude of the triangle end text A B C text . Prove that end text A C left parenthesis A D minus D C right parenthesis equals A B squared minus B C squared$

hint Hint:

Hint :- Applying pythagoras theorem to both triangles ABD and BDC. find the BD2 from both equations .Equate both the equations to get the required condition .

The correct answer is: Hence proved

Aim :- Prove that $A C left parenthesis A D minus D C right parenthesis equals A B squared minus B C squared$
Explanation(proof ) :-
Applying pythagoras theorem in ΔBCD ,We get
$B C squared equals B D squared plus C D squared$ — Eq1
Applying pythagoras theorem in ΔABD ,We get
$A B squared equals B D squared plus A D squared not stretchy rightwards double arrow B D squared equals A B squared minus A D squared$ —Eq2
Substitute Eq1 in Eq2 ,
$B C squared equals A B squared minus A D squared plus C D squared not stretchy rightwards double arrow B C squared equals A B squared plus C D squared minus A D squared$

$text By using end text a squared minus b squared equals left parenthesis a minus b right parenthesis left parenthesis a plus b right parenthesis text and end text A C equals A D plus C D text we get, end text$
$not stretchy rightwards double arrow B C squared equals A B squared plus left parenthesis A D plus C D right parenthesis left parenthesis C D minus A D right parenthesis$
$B C squared equals A B squared plus left parenthesis A C right parenthesis left parenthesis C D minus A D right parenthesis$
$negative A C left parenthesis C D minus A D right parenthesis equals A B squared minus B C squared$
$therefore A C left parenthesis A D minus D C right parenthesis equals A B squared minus B C squared$
Hence proved

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