In the picture, a line through the centre of a circle meets a chord of the circle: What are the lengths of the two pieces of the chord?

From the figure: OA = OD = OE = 7 cm (Radius) PE = OE - OP = 7 - 3 = 4 cm . The intersection may be with in the circle. So, internal bisector theorem of chords C P × P B = P E × P D ⇒ C P × P B = 4 × ( 7 + 3 ) = 4 × 10 = 40 CP × PB = 40 … … … … … (i) CP + PB = 13 ....... (ii) From (i) and (ii), ⇒ ( 13 – P B ) P B = 40 By solving the above equation: ⇒ 13 P B - P B 2 - 40 = 0 ⇒ P B 2 - 13 P B + 40 = 0 ⇒ P B 2 - 8 P B - 5 P B + 40 = 0 ⇒ P B - 8 P B - 5 = 0 ∴ P B = 5 , 8 In the figure, P B > P C . P B = 8 c m , P C = 5 c m . Length of two pieces of chord are 8 cm, 5 cm .