AB and CD are two chords of a circle intersecting at point P inside the circle. If

(i) AB = 24 cm, AP = 4 cm and PD = 8 cm, determine CP.

(ii) AP = 3 cm, PB = 2.5 cm and CD = 6.5 cm, determine CP.

(i) From figure,

⇒ AB = AP + PB

⇒ 24 = 4 + PB

⇒ PB = 24 - 4

⇒ PB = 20 cm.

We know that,

If two chords intersect in a circle, then the products of the measures of the segments of the chords are equal.

⇒ AP × PB = CP × PD

⇒ 4 × 20 = CP × 8

⇒ 80 = 8CP

⇒ CP = $\dfrac{80}{8}$ = 10.

Hence, CP = 10 cm.

(ii) Let CP = x, so PD = (6.5 - x).

We know that,

If two chords intersect in a circle , then the products of the measures of the segments of the chords are equal.

⇒ AP × PB = CP × PD

⇒ 3 × 2.5 = x(6.5 - x)

⇒ 7.5 = 6.5x - x²

⇒ x² - 6.5x + 7.5 = 0

⇒ x² - 5x - 1.5x + 7.5 = 0

⇒ x(x - 5) - 1.5(x - 5) = 0

⇒ (x - 1.5)(x - 5) = 0

⇒ x - 1.5 = 0 or x - 5 = 0

⇒ x = 1.5 or x = 5.

Hence, CP = 1.5 cm or 5 cm.