Question

AB and CD are two chords of a circle intersecting at a point P outside the circle when produced
such that PA = 16 cm, PC = 10 cm and PD = 8 cm. Find AB.
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Answer 1

two chords of a circle intersect each other internally or externally, then area of rectangle contained by the segment of one chord is equal to the area of rectangle contained by the segment of other chord.

Using above theorem,

PA × PB = PC × PD …(1)

(i) Given: AB = 4 cm

BP = 5 cm

PD = 3 cm

∵ AP = AB + BP

⇒ AP = 4 cm + 5 cm

⇒ AP = 9 cm

Putting the values in (1),

9 cm × 5 cm = PC × 3 cm

⇒ PC = 15 cm

∵ PC = CD + DP

⇒ 15 cm = CD + 3 cm

⇒ CD = 15 cm – 3 cm

⇒ CD = 12 cm

CD = 12 cmHence, CD = 12 cm

CD = 12 cmHence, CD = 12 cm(ii) Given: BP = 3 cm

CD = 12 cmHence, CD = 12 cm(ii) Given: BP = 3 cmCP = 6 cm

CD = 12 cmHence, CD = 12 cm(ii) Given: BP = 3 cmCP = 6 cmCD = 2 cm

CD = 12 cmHence, CD = 12 cm(ii) Given: BP = 3 cmCP = 6 cmCD = 2 cm∵ CP = CD + DP

CD = 12 cmHence, CD = 12 cm(ii) Given: BP = 3 cmCP = 6 cmCD = 2 cm∵ CP = CD + DP⇒ DP = CP – CD

CD = 12 cmHence, CD = 12 cm(ii) Given: BP = 3 cmCP = 6 cmCD = 2 cm∵ CP = CD + DP⇒ DP = CP – CD⇒ DP = 6 cm – 2 cm

CD = 12 cmHence, CD = 12 cm(ii) Given: BP = 3 cmCP = 6 cmCD = 2 cm∵ CP = CD + DP⇒ DP = CP – CD⇒ DP = 6 cm – 2 cm⇒ DP = 4 cm

CD = 12 cmHence, CD = 12 cm(ii) Given: BP = 3 cmCP = 6 cmCD = 2 cm∵ CP = CD + DP⇒ DP = CP – CD⇒ DP = 6 cm – 2 cm⇒ DP = 4 cmPutting the values in (1),

PA × 3 cm = PC × PD

⇒ PA × 3 cm = 6 cm × 4 cm

⇒ PA = 8 cm

∵ AP = AB + BP

⇒ 8 cm = AB + 3 cm

⇒ AB = 8 cm – 3 cm

⇒ AB = 5 cm

Hence, AB = 5 cm

Answer 2

Correct Question :-- AB and CD are two chords of a circle intersecting at a point P outside the circle when produced

such that PA = 16 cm, PC = 10 cm and PD = 8 cm. Find AB . ?

Concept used :---

Intersecting Secants Theorem:-

When two secant lines intersect each other outside a circle, the products of their segments are equal.

or, we can say that, When two secant lines AB and CD intersect outside the circle at a point P, then

PA.PB = PC.PD

_____________________________

Given :--

→ PA = 16cm

→ PC = 10cm

→ PD = 8cm .

Putting all values now, we get,

→ 16 × PB = 10 × 8

→ 16 × PB = 80

→ PB = 80/16

→ PB = 5 cm .

____________________________

Now,

∵ PA = AB + BP

⇒ 16 cm = AB + 5 cm

⇒ AB = 16 cm – 5 cm

⇒ AB = 11 cm

Hence, Length of AB is 11 cm......