A chord

P Q
whose length is

4.8 cm
, of a circle of radius

3 cm
. The tangents at

P
and

Q
intersect at a point

T
as shown in the figure then the length of

TP
is ____cm.

Question

A chord

P Q
   whose length is

4.8 cm
  , of a circle of radius

3 cm
  . The tangents at

P
  and

Q
  intersect at a point

T
  as shown in the figure then the length of

TP
   is ____cm.

Accepted Answer

A chord

P Q
   whose length is

4.8 cm
  , of a circle of radius

3 cm
  . The tangents at

P
  and

Q
  intersect at a point

T
  as shown in the figure then the length of

TP
   is 4cm.Given that,

PQ=4.8 cm

OP=3 cm

Joining

O
   and

Q
  ,Equating between

∠PTO
   and

∠OTQ
  ,

OP=OQ
   [Radii of the same circle]

PT=PQ
  [Tangents from a same external point]

OT
  is the common side,Therefore by SSS congruency

ΔPTO
  and

ΔOTQ
   are congruentAnd

∠PTO=∠OTQ
   …..i)Calculating the value of

PR
  using (i),

∠PTO=∠OTQ

PT=PQ

[Tangents subtended by common external point]

RT
  is a common side,Therefore by SAS congruency rule

ΔPRT
   and

ΔRTQ 
  are congruent.

PR=RQ=
      1
      2
     
     PQ

PR=PQ=2.4 cm

Calculating the values of

TR
   and

OR
  ,Applying Pythagoras theorem on

ΔPRO
   and using equation ii),

O
      P
      2
     
     =P
      R
      2
     
     +O
      R
      2

3
      2
     
     =
      2.4
      2
     
     +O
      R
      2

OR=
      
       9−5.76

OR=
      
       3.24

OR=1.8.....iii)

Applying Pythagoras theorem on

ΔOPT
  ,

O
      P
      2
     
     +P
      T
      2
     
     =O
      T
      2

3
      2
     
     +
      
       T
        R
        2
       
       +P
        R
        2

=
      (TR+OR)
      2

9+5.76+T
      R
      2
     
     =T
      R
      2
     
     +3.24+2×1.8×TR

11.52=3.6TR

TR=3.2.....iv)

Calculating the value of

PT
  ,Again applying the Pythagoras theorem on

ΔPRT
   and using equation iv),

P
      T
      2
     
     =T
      R
      2
     
     +P
      R
      2

P
      T
      2
     
     =
      3.2
      2
     
     +
      2.4
      2

P
      T
      2
     
     =16

PT=4

Thus the length of

PT
  is

4 cm
  .

A chord $P Q$ whose length is $4.8 cm$ , of a circle of radius $3 cm$ . The tangents at $P$ and $Q$ intersect at a point $T$ as shown in the figure then the length of $T P$ is ____cm.

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A chord P Q whose length is 4.8 cm , of a circle of radius 3 cm . The tangents at P and Q intersect at a point T as shown in the figure then the length of TP is ____cm.