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Maths-

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Question

$text In a Triangle end text A B C comma less than B greater than 90 to the power of ring operator text and end text A D text is drawn perpendicular to end text B C text . Prove that end text A C squared equals A B squared plus B C squared plus 2 B C. B D$

hint Hint:

Hint :- Applying pythagoras theorem to both triangles ABD and ADC. find the AC2 from equations .Equate both the equations to get the required condition .

The correct answer is: Hence proved

Aim :- Prove that $A C squared equals A B squared plus B C squared plus 2 B C times B D$
Explanation(proof ) :-
As angle B is Obtuse the perpendicular line AD lies outside the triangle
Applying pythagoras theorem in ΔABD ,We get
$A B squared equals B D squared plus A D squared$ —Eq1
As angle b is acute the perpendicular line AD lies inside the triangle
Applying pythagoras theorem in ΔACD ,We get
$A C squared equals A D squared plus C D squared left square bracket text Substitute end text CD equals BD plus BC text from diagram end text right square bracket$
$A C squared equals A D squared plus left parenthesis B D plus B C right parenthesis squared open square brackets left parenthesis a plus b right parenthesis squared equals a squared plus b squared plus 2 a b close square brackets$
$A C squared equals A D squared plus open parentheses B D squared plus B C squared plus 2 B D times B C close parentheses$
$A C squared equals open parentheses A D squared plus B D squared close parentheses plus B C squared plus 2 B D times B C$
Substitute Eq 1 $open parentheses A B squared equals B D squared plus A D squared close parentheses$ in the above equation
We get , $A C squared equals A B squared plus B C squared plus 2 B C times B D$
Hence proved

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