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.

Infinity Learn · 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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Detailed Solution

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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.

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Detailed Solution

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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 $T P$ is ____cm.