In the given figure,

∠ACB
=90° and

CD⊥AB
. State if the given condition true or false “

C
B
2

C
A
2

=

BD

AD

”.

Question

In the given figure,

∠ACB
  =90° and

CD⊥AB
  . State if the given condition true or false “

C
      B
      2

C
      A
      2

=
    
     BD
    
     AD

”.

Accepted Answer

It is given that, ∠ACB=90° and CD⊥AB.If a perpendicular is drawn from the right angle's vertex to the hypotenuse, the triangle on both sides of the perpendicular is similar to the whole triangle and to each other.We can observe from the figure that in triangle ACD and ABC, ∠ADC=∠ACB(90°)∠CAD=∠CAB(same)Triangle ACD ≈ΔABC by AA criterion.In triangle BCD and BAC∠BDC=∠BCA(90°)∠CBD=∠CBA(same)Hence, the triangle BCD≈BAC by AA criterion.We can observe from the figure that, ΔACD~ΔABC and ΔBCD~ΔBAC with AA similarity criterion. Since, corresponding sides of similar triangles are proportional, we can write that,

ΔACD∼ΔABC

⇒
      
       AC
      
       AB
     
     =
      
       AD
      
       AC

⇒A
      C
      2
     
     =AB×AD….(1)

ΔBCD∼ΔBAC

⇒
      
       BC
      
       BA
     
     =
      
       BD
      
       BC

⇒B
      C
      2
     
     =BA×BD…(2)

Dividing (2) by (1),

B
        C
        2

A
        C
        2

=
      
       BA×BD
      
       AB×AD

⇒
      
       B
        C
        2

A
        C
        2

=
      
       BD
      
       AD

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

In the given figure, $∠ A C B$ =90° and $C D ⊥ A B$ . State if the given condition true or false “ $C B 2 C A 2 = B D A D$ ”.

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In the given figure, ∠ACB =90° and CD⊥AB . State if the given condition true or false “ C B 2 C A 2 = BD AD ”.