In the given figure,

D
is the midpoint of side BC and

AE⊥BC
. If

BC=a,

AC=b,

AB=c,
AE=h, DE=x and

AD=p
. State if the given equation is true or false

b
2

=
p
2

+ax+

a
2

4

.

Question

In the given figure,

D
  is the midpoint of side BC and

AE⊥BC
  . If

BC=a,

AC=b,

AB=c,
   AE=h, DE=x and

AD=p
  . State if the given equation is true or false

b
    2
   
   =
    p
    2
   
   +ax+

a
      2

4

.

Accepted Answer

The question states that,

AE⊥BC

BD=CD

BC=a,AB=c,AD=p,AC=b,DE=x,AE=h

Applying Pythagoras theorem for

ΔABC
   .

ΔAEC
   Applying Pythagoras theorem in

ΔAEC

A
      C
      2
     
     =A
      E
      2
     
     +C
      E
      2

b
      2
     
     =
      h
      2
     
     +

BC
         2
        
        +ED
      
      2

b
      2
     
     =
      h
      2
     
     +

a
        2
       
       +
        x
        2

b
      2
     
     =
      h
      2
     
     +

a
        2

4
     
     +
      x
      2
     
     +ax     ​​[​​ eq.2 ​​]​​

In

ΔAED

A
      D
      2
     
     =A
      E
      2
     
     +D
      E
      2

p
      2
     
     =
      h
      2
     
     +
      x
      2
     
           ​​[​​ eq.3 ​​]​​

Using eq. (3) in eq. (1),

c
      2
     
     =
      h
      2
     
     +
      x
      2
     
     +

a
        2

4
     
     −ax

c
      2
     
     =
      p
      2
     
     −ax+

a
        2

4
     
           ​​[​​ eq.4 ​​]​​

Using eq. (3) in (2), we can write that,

b
      2
     
     =
      h
      2
     
     +
      x
      2
     
     +

a
        2

4+ax

b
      2
     
     =
      p
      2
     
     +ax+

a
        2

4

Therefore, the given statement is true i.e., .Hence, the correct option is 2.

In the given figure, $D$ is the midpoint of side BC and $A E ⊥ B C$ . If $BC = a,$ $AC = b,$ $AB = c,$ AE=h, DE=x and $A D = p$ . State if the given equation is true or false $b 2 = p 2 + a x + a 2 4$ .

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In the given figure, D is the midpoint of side BC and AE⊥BC . If BC=a, AC=b, AB=c, AE=h, DE=x and AD=p . State if the given equation is true or false b 2 = p 2 +ax+ a 2 4 .

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In the given figure, $D$ is the midpoint of side BC and $A E ⊥ B C$ . If $BC = a,$ $AC = b,$ $AB = c,$ AE=h, DE=x and $A D = p$ . State if the given equation is true or false $b 2 = p 2 + a x + a 2 4$ .