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Question 5
At one end A of a diameter AB of a circle of radius 5 cm, tangent XAY is drawn to the circle. The length of the chord CD parallel to XY and at a distance 8 cm from A, is

(A) 4 cm
(B) 5 cm
(C) 6 cm
(D) 8 cm

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Solution

First draw a circle of radius 5 cm having centre O. A tangent XY is drawn at point A

A Chord CD is drawn which is parallel to XY and at a distance of 8 cm from A.
Now , OAY=90
[Tangent at any point of a circle is perpendicular to the radius through the point of contact]
OAY+OED=180 [sum of cointerior angle is 180] OED=90Also, AE=8cm.Join OCNow, in right angled ΔOEC.,OC2=OE2+EC2[by Pythagoras theorem]EC2=OC2OE2 =5232[OC=radius=5cm,OE=AEAO=85=3cm]=259=16EC=4cm
Hence, length of chord CD = 2 CE = 2 × 4 = 8 cm
[Since perpendicular from centre to the chord bisects the chord]

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