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∘[sumofcointeriorangleis180∘]⇒∠OED=90∘Also,AE=8cm.JoinOCNow,inrightangledΔOEC.,OC2=OE2+EC2[byPythagorastheorem]⇒EC2=OC2−OE2=52−32[∵OC=radius=5cm,OE=AE−AO=8−5=3cm]=25−9=16⇒EC=4cm
Hence, length of chord CD = 2 CE = 2 × 4 = 8 cm
[Since perpendicular from centre to the chord bisects the chord]