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 of 8 cm from A is ______

#### Options

4 cm

5 cm

6 cm

8 cm

#### Solution

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 of 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,

OC^{2} = OE^{2} + EC^{2} ......[By Pythagoras theorem]

⇒ EC^{2} = OC^{2} – OE^{2}

⇒ EC^{2} = 5^{2} – 3^{2}

⇒ EC^{2} = 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