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Diagram:
Calculation:
AB and AC are two equal chords of a circle, therefore the centre of the circle lies on the bisector of ∠BAC.
⇒ OA is the bisector of ∠BAC.
Again, the internal bisector of an angle divides the opposite sides in the ratio of the sides containing the angle.
P divides BC in the ratio = 6 : 6 = 1 : 1.
⇒ P is mid-point of BC.
⇒ OP ⊥ BC.
In ΔABP, by Pythagoras theorem,
AB2 = AP2 + BP2
⇒ BP2 = 62 - AP2 ---- (1)
In right triangle OBP, we have
OB2 = OP2 + BP2
⇒ 52 = (5 - AP)2 + BP2
⇒ BP2 = 25 - (5 - AP)2 ---- (2)
Equating (1) and (2), we get
62 - AP2 = 25 - (5 - AP)2
⇒ 11 - AP2 = -25 - AP2 + 10AP
⇒ 36 = 10AP
⇒ AP = 3.6 cm
Putting AP in (1), we get
BP2 = 62 - (3.6)2 = 23.04
⇒ BP = 4.8 cm
⇒ BC = 2BP = 2 × 4.8 = 9.6 cm = length of chord
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