Question

2. In the following figure (not to scale), two chords
XY and PQ are intersecting at the point A. The line
segment joining X and P is a diameter of the circle,
XAP= 120° and XY = PQ = 18 cm. Find the dis-
tance between the centre of the circle and the point A.


(a) 3 cm
(c) 8 cm
(b) 4 cm
(d) 6 cm​

Answer 1

Given :   two chords  XY and PQ are intersecting at the point A. The line

segment joining X and P is a diameter of the circle,  XAP= 120° and XY = PQ = 18 cm.  

To find :  distance between the centre of the circle and the point A.

Solution:

XP is a diameter

=> ∠XYP = ∠XQP  = 90°

XY = PQ =  18 cm

XP = XP  ( coomon)

=> ΔXYP ≅ ΔPQX

XQ = PY

ΔXAQ ≅ ΔPAY

=> XA = PA

=> ∠AXP = ∠APX

∠XAP = 120°

=> ∠AXP = ∠APX = (180° - 120°)/2 = 30°

in ΔXPY

Sin 30 = XY/XP

=> √3/2 = 18/XP

=> XP = 12√3  cm

=> XO = OP = XP/2  = 6√3  cm

in Δ XAO

tan 30°  =  AO/XO

=> 1/√3  = AO /  6√3  

=> AO = 6  cm

distance between the centre of the circle and the point A. = 6 cm

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