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Maths-

General

Easy

Question

In the figure, ABC is a triangle in which $A D perpendicular B C$ . Show that $A C squared equals A B squared plus B C squared minus 2 B C times B D$
.

The correct answer is: Hence proved

Solution :-
Aim :- Prove that $A C squared equals A B squared plus B C squared minus 2 B C times B D$
Hint :- Applying pythagoras theorem to both triangles ABD and ADC. find the AD² from both equations .Equate both the equations to get the required condition .
Explanation(proof ) :-
Applying pythagoras theorem in ΔACD ,We get
$A C squared equals A D squared plus C D squared minus Eq 1$
Applying pythagoras theorem in ΔABD ,We get
$A B squared equals B D squared plus A D squared not stretchy rightwards double arrow A D squared equals A B squared minus B D squared minus Eq 2$
Substitute Eq2 in Eq1 ,
$A C squared equals A B squared minus B D squared plus C D squared not stretchy rightwards double arrow A C squared equals A B squared plus C D squared minus B D squared$
By using $a squared minus b squared equals left parenthesis a minus b right parenthesis left parenthesis a plus b right parenthesis text and end text B C equals B D plus C D text we get, end text$
$not stretchy rightwards double arrow A C squared equals A B squared plus left parenthesis B D plus C D right parenthesis left parenthesis C D minus B D right parenthesis$
$not stretchy rightwards double arrow A C squared equals A B squared plus B C left parenthesis C D plus B D minus 2 B D right parenthesis left parenthesis text Add end text BD text and subtract end text BD right parenthesis$
$not stretchy rightwards double arrow A C squared equals A B squared plus B C left parenthesis B C minus 2 B D right parenthesis$
$therefore A C squared equals A B squared plus B C squared minus 2 B C times B D$
Hence proved

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