Question

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 ______ 

Options

• 4 cm

• 5 cm

• 6 cm

• 8 cm

Answer 1

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 8 cm.

Explanation:

First, draw a circle of radius 5 cm with 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 circle is perpendicular to the radius through the point of contact]

∠OAY + ∠OED = 180°   ...[∵ Sum of cointerior angles is 180°]

⇒ ∠OED = 180° – 90° = 90°

Also, AE = 8 cm.

Join OC

OC = 5 cm  ...[Radius of circle]

OE = AE – OA

= 8 – 5

= 3 cm

Now, in right angled ∆OEC,

OC² = OE² + EC²   ...[By Pythagoras theorem]

⇒ EC² = OC² – OE²

⇒ EC² = 5² – 3²

⇒ EC² = 25 – 9 = 16

⇒ EC = 4 cm

Since, perpendicular from centre to the chord bisects the chord.

∴ CE = ED

⇒ CD = 2 × EC

⇒ CD = 2 × 4

⇒ CD = 8 cm