PQ
is a chord of length

16 cm
of a circle of radius

10 cm
. If the tangents at

P
and

Q
intersect at a point

T
, then the length of

TP
will be ____ cm.

Question

PQ
  is a chord of length

16 cm
  of a circle of radius

10 cm
  . If the tangents at

P
  and

Q
  intersect at a point

T
  , then the length of

TP
  will be ____ cm.

Accepted Answer

PQ
  is a chord of length

16 cm
  of a circle of radius

10 cm
  . If the tangents at

P
  and

Q
  intersect at a point

T
  , then the length of

TP
  will be 10.67cm.We must know the Perpendicular Bisector Pythagoras theorem where the hypotenuse is the longest side, as it is opposite to the angle of 90°. Pythagoras Theorem formula can be written as,

Hypotenuse 
    2
   
   =
    
      Perpendicular 
    2
   
   +
    
      Base 
    2

Given that

PQ
  is a chord which is

16 cm
  and radius is

10 cm
  .Assuming that,

TR = y

TP = x
  . The perpendicular drawn from the center to the chord bisects it,

PR

=PQ

PR+RQ

=16

PR+PR

=16

PR

=8

In

ΔPOR
  ,

PO
        2

=
        
         OR
        2
       
       +
        
         PR
        2

10
        2

=
        
         OR
        2
       
       +
        8
        2

OR
        2

=36

OR

=6

Using the Pythagoras Theorem in right angled

ΔTPR
  ,

T
        P
        2

=T
        R
        2
       
       +P
        R
        2

x
        2

=
        y
        2
       
       +
        8
        2

x
        2

=
        y
        2
       
       +64.....i)

Using the Pythagoras Theorem in right angled

ΔTPQ
  ,

TO
        2

=
        
         TP
        2
       
       +
        
         PO
        2

(y+6)
        2

=
        x
        2
       
       +
        
         10
        2

y
        2
       
       +12y+36

=
        x
        2
       
       +100

y
        2
       
       +12y

=
        x
        2
       
       +64.....ii)

Solving the equations (i) and (ii), we get,

x=10.67

TP=10.67 cm

Thus the length of

TP
  is

10.67 cm
  .

$PQ$ is a chord of length $16 cm$ of a circle of radius $10 cm$ . If the tangents at $P$ and $Q$ intersect at a point $T$ , then the length of $TP$ will be ____ cm.

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PQ is a chord of length 16 cm of a circle of radius 10 cm . If the tangents at P and Q intersect at a point T , then the length of TP will be ____ cm.