State if the following is true or false.In ΔABC, angleB = 90° and D is the midpoint of BC, then

A
C
2

=A
D
2

+3C
D
2

.

Question

State if the following is true or false.In ΔABC, angleB = 90° and D is the midpoint of BC, then

A
    C
    2
   
   =A
    D
    2
   
   +3C
    D
    2
   
   .

Accepted Answer

Consider the figure,The Pythagoras Theorem is given as,

Hypotenuse
     
    2
   
   =

Base
     
    2
   
   +

Perpendicular
     
    2

Using the Pythagoras theorem,

AB
      
      2
     
     +

BC
      
      2
     
     =

AC
      
      2

⇒

AB
      
      2
     
     =

AC
      
      2
     
     −

BC
      
      2

Again,

AB
      
      2
     
     +

BD
      
      2
     
     =

AD
      
      2

⇒

AB
      
      2
     
     =

AD
      
      2
     
     −

BD
      
      2

Equating the right side of the equations

AB
     
    2
   
   =

AC
     
    2
   
   −

BC
     
    2

and

AB
     
    2
   
   =

AD
     
    2
   
   −

BD
     
    2

to obtain an equation,

AC
      
      2
     
     −

BC
      
      2
     
     =

AD
      
      2
     
     −

BD
      
      2

⇒

AC
      
      2
     
     =

AD
      
      2
     
     −

BD
      
      2
     
     +

BC
      
      2

……(1)As D is the mid-point of BC. So,

BC=2CD
  Substitute CD for BD and 2CD for BC in

AC
     
    2
   
   =

AD
     
    2
   
   −

BD
     
    2
   
   +

BC
     
    2

to obtain the required equation,

AC
      
      2
     
     =

AD
      
      2
     
     −

CD
      
      2
     
     +

2CD
      
      2

⇒

AC
      
      2
     
     =

AD
      
      2
     
     −

CD
      
      2
     
     +4

CD
      
      2

⇒

AC
      
      2
     
     =

AD
      
      2
     
     +3

CD
      
      2

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

State if the following is true or false.

In ΔABC, angleB = 90° and D is the midpoint of BC, then $A C 2 = A D 2 + 3 C D 2 .$

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State if the following is true or false.In ΔABC, angleB = 90° and D is the midpoint of BC, then A C 2 =A D 2 +3C D 2 .

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State if the following is true or false.

In ΔABC, angleB = 90° and D is the midpoint of BC, then $A C 2 = A D 2 + 3 C D 2 .$