Two chords AB, CD of a circle intersect internally at a point P. If
(i) AP = 6 cm, PB = 4 cm and PD = 3 cm, find PC.
(ii) AB = 12 cm, AP = 2 cm, PC = 5 cm, find PD.
(iii) AP = 5 cm, PB = 6 cm and CD = 13 cm, find CP.

Question

Two chords AB, CD of a circle intersect internally at a point P. If
(i) AP = 6 cm, PB = 4 cm and PD = 3 cm, find PC.
(ii) AB = 12 cm, AP = 2 cm, PC = 5 cm, find PD.
(iii) AP = 5 cm, PB = 6 cm and CD = 13 cm, find CP.

KnowledgeBoat · Accepted Answer

We know that when two chords of a circle intersect internally or externally, then the products of the lengths of segments are equal. Given, chords AB and CD of a circle intersect internally at a point P. So, PA.PB = PC.PD (i) Given, AP = 6 cm, PB = 4 cm and PD = 3 cm. We know that, PA.PB = PC.PD ⇒ 6 × 4 = PC × 3 ⇒ 24 = 3PC ⇒ PC = = 8 cm. Hence, the length of PC = 8 cm. (ii) Given, AB = 12 cm, AP = 2 cm, PC = 5 cm PB = AB - AP = 12 - 2 = 10 cm. We know that, PA.PB = PC.PD ⇒ 2 × 10 = 5 × PD ⇒ 20 = 5PD ⇒ PD = = 4 cm. Hence, the length of PD = 4 cm. (iii) Given, AP = 5 cm, PB = 6 cm and CD = 13 cm Let PC = x, so PD = 13 - x We know that, PA.PB = PC.PD ⇒ 5 × 6 = x(13 - x) ⇒ 30 = 13x - x² ⇒ x² - 13x + 30 = 0 ⇒ x² - 10x - 3x + 30 = 0 ⇒ x(x - 10) - 3(x - 10) = 0 ⇒ (x - 3)(x - 10) = 0 ⇒ x - 3 = 0 or x - 10 = 0 ⇒ x = 3 or x = 10. Hence, PC = 3 cm or 10 cm.

Mathematics

Two chords AB, CD of a circle intersect internally at a point P. If
(i) AP = 6 cm, PB = 4 cm and PD = 3 cm, find PC.
(ii) AB = 12 cm, AP = 2 cm, PC = 5 cm, find PD.
(iii) AP = 5 cm, PB = 6 cm and CD = 13 cm, find CP.

Circles

ICSE

Related Questions

In the figure (ii) given below, O is the center of the circle and PT is the tangent to the circle at P. Given ∠QPT = 30°, calculate
(i) ∠PRQ
(ii) ∠POQ.

In the figure (i) given below, PT is a tangent to the circle. Find TP if AT = 16 cm and AB = 12 cm.

In the figure (i) given below, O is the centre of the circumcircle of triangle XYZ. Tangents at X and Y intersect at T. Given ∠XTY = 80° and ∠XOZ = 140°, calculate the value of ∠ZXY.

In the figure (ii) given below, diameter AB and chord CD of a circle meet at P. PT is a tangent to the circle at T. CD = 7.8 cm, PD = 5 cm, PB = 4 cm. Find :
(i) AB
(ii) the length of tangent PT.

Mathematics

Two chords AB, CD of a circle intersect internally at a point P. If(i) AP = 6 cm, PB = 4 cm and PD = 3 cm, find PC.(ii) AB = 12 cm, AP = 2 cm, PC = 5 cm, find PD.(iii) AP = 5 cm, PB = 6 cm and CD = 13 cm, find CP.

Circles

ICSE

Related Questions

In the figure (ii) given below, O is the center of the circle and PT is the tangent to the circle at P. Given ∠QPT = 30°, calculate(i) ∠PRQ(ii) ∠POQ.

In the figure (i) given below, PT is a tangent to the circle. Find TP if AT = 16 cm and AB = 12 cm.

In the figure (i) given below, O is the centre of the circumcircle of triangle XYZ. Tangents at X and Y intersect at T. Given ∠XTY = 80° and ∠XOZ = 140°, calculate the value of ∠ZXY.

In the figure (ii) given below, diameter AB and chord CD of a circle meet at P. PT is a tangent to the circle at T. CD = 7.8 cm, PD = 5 cm, PB = 4 cm. Find :(i) AB(ii) the length of tangent PT.

Two chords AB, CD of a circle intersect internally at a point P. If
(i) AP = 6 cm, PB = 4 cm and PD = 3 cm, find PC.
(ii) AB = 12 cm, AP = 2 cm, PC = 5 cm, find PD.
(iii) AP = 5 cm, PB = 6 cm and CD = 13 cm, find CP.

In the figure (ii) given below, O is the center of the circle and PT is the tangent to the circle at P. Given ∠QPT = 30°, calculate
(i) ∠PRQ
(ii) ∠POQ.

In the figure (ii) given below, diameter AB and chord CD of a circle meet at P. PT is a tangent to the circle at T. CD = 7.8 cm, PD = 5 cm, PB = 4 cm. Find :
(i) AB
(ii) the length of tangent PT.