State if the following is true or false.

P
and Q are the mid-points of the sides CAand CB respectively of

ΔABC
right angled at

C.
then

4

AQ
2

+

BP
2

=5

AB
2

.

Question

State if the following is true or false.

P
  and Q are the mid-points of the sides CAand CB respectively of

ΔABC
   right angled at

C.
  then

4

AQ
      2
     
     +
      
       BP
      2

=5
    
     AB
    2

.

Accepted Answer

We need to check whether

4

AQ
      2
     
     +
      
       BP
      2

=5
    
     AB
    2

holds good or not.Consider the figure,Here, ABC is the triangle and P and Q are the points on the line’s AC and BC respectively. In triangle ACB,

∠c=
    
     90
    °

AB2=AC2+BC2........(1)Now apply the Pythagoras theorem in ACQ,

A
      Q
      2
     
     =A
      C
      2
     
     +Q
      C
      2

⇒A
      Q
      2
     
     =A
      C
      2
     
     +

BC
         2

2

⇒A
      Q
      2
     
     =A
      C
      2
     
     +

BC
       4

2

⇒4A
      Q
      2
     
     =4A
      C
      2
     
     +B
      C
      2
     
     .......(2)

Using Pythagoras theorem in

ΔPCB
  ,

P
      B
      2
     
     =P
      C
      2
     
     +B
      C
      2

⇒P
      B
      2
     
     =
      
       A
        C
        2

4
     
     +B
      C
      2

⇒4P
      B
      2
     
     =A
      C
      2
     
     +4B
      C
      2
     
     ....(3)

Add equation (2) and (3),

4A
    Q
    2
   
   +4P
    B
    2
   
   =4A
    C
    2
   
   +B
    C
    2
   
   +A
    C
    2
   
   +4B
    C
    2

⇒4
    
     A
      Q
      2
     
     +P
      B
      2

=5
    
     A
      C
      2
     
     +B
      C
      2

Substitute the value for

A
      C
      2
     
     +B
      C
      2

by using equation (i),

4
    
     A
      Q
      2
     
     +P
      B
      2

=5
    
     A
      B
      2

The identity

4
    
     A
      Q
      2
     
     +P
      B
      2

=5
    
     A
      B
      2

holds good.Therefore, the statement is true.Hence, option 1 is correct.

State if the following is true or false.

$P$ and Q are the mid-points of the sides CAand CB respectively of $Δ ABC$ right angled at $C .$ then $4 AQ 2 + BP 2 = 5 AB 2$ .

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State if the following is true or false. P and Q are the mid-points of the sides CAand CB respectively of ΔABC right angled at C. then 4 AQ 2 + BP 2 =5 AB 2 .