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  • In ABC, if∠B = 90° and D is the midpoint of BC, then AC2 = AD2 + CD2. (True/False) Option: 1</s

In ABC, if∠B = 90° and D is the midpoint of BC, then AC2 = AD2 + CD2. (True/False) 

Option: 1

True


Option: 2

False


Answers (1)

best_answer

Given that

∠B = 90° and D is the midpoint of BC,

So, AD is the median

Now, from the Appolonius Theorem

A B^{2}+A C^{2}=2\left(A D^{2}+D C^{2}\right)

A C^{2}=2\left(A D^{2}+D C^{2}\right)-A B^{2}

ABD is also a right angle triangle, 

So, AB^2=AD^2-BD^2=AD^2-CD^2\;\;\;\;\;\;(\because BD=CD)

From the above

\\\Rightarrow A C^{2}=2\left(A D^{2}+D C^{2}\right)-A B^{2}\\\Rightarrow A C^{2}=2\left(A D^{2}+D C^{2}\right)-\left (A D^{2}-CD^2 \right )\\\mathrm{\;\;\;\;\;\;\;\;}\;\;\;=AD^2+3CD^2

Hence, the statement is true.

 

Posted by

Pankaj

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