Question

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

Answer 1

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 , $\angle OAY={90}^{\circ }$
[Tangent at any point of a circle is perpendicular to the radius through the point of contact]
$\angle OAY+\angle OED={180}^{\circ }\left[sumofcointeriorangleis{180}^{\circ }\right]\Rightarrow \angle OED={90}^{\circ }Also,AE=8cm.JoinOCNow,inrightangled\Delta OEC.,O{C}^2=O{E}^2+E{C}^2\left[byPythagorastheorem\right]\Rightarrow E{C}^2=O{C}^2-O{E}^2=5^2-3^2[\because OC=radius=5cm,OE=AE-AO=8-5=3cm]=25-9=16\Rightarrow EC=4cm$
Hence, length of chord CD = 2 CE = 2 $\times$ 4 = 8 cm
[Since perpendicular from centre to the chord bisects the chord]