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
4cm
B
5cm
C
6m
D
8cm

Solution:

Given that, at one end A of a diameter AB of a circle of radius 5 cm, tangent XAY is drawn to the circle.

We have to find the length of the chord CD parallel to XY.
As we know that tangent at any point of a circle is perpendicular to the radius at the point of contact.

∠ O A Y = 90 °

As sum of co-interior angle is 180°. We get,

∠ O A Y + Δ O E D = 180 °

∠ O E D = 90 °, A E = 8 c m [g i v e n]

Now in △OEC, by Pythagoras theorem,

O C 2 = O E 2 + E C 2 ⇒ E C 2 = O C 2 − O E 2 ⇒ E C 2 = 52 − 32 ⇒ E C 2 = 25 − 9 ⇒ E C 2 = 16 ⇒ E C = 4 c m

Therefore, length of chord CD = 2×CE (∵perpendicular from center of the circle to the chord bisects the chord)
⇒CD=2×4=8cm
∴The length of the chord CD is 8cm.
Hence, option 4 is correct.