Question

Given O is centre of the circle with chord AB = 8 cm. OA = 5 cm and OD ⊥ AB. The length of CD is ______.

विकल्प

• 3 cm

• 5 cm

• 2 cm

• none of these

Answer 1

Given O is centre of the circle with chord AB = 8 cm. OA = 5 cm and OD ⊥ AB. The length of CD is 2 cm.

Explaantion:

Since OD ⊥ AB, the perpendicular from the centre of a circle to a chord bisects the chord. Given that the total length of chord AB is 8 cm, the length of segment AC is half of AB:

AC = `8/2`

∴ AC = 4 cm

In the right-angled triangle ΔACO (where ∠ACO = 90°), the radius OA serves as the hypotenuse with length 5 cm, and AC = 4 cm. Using the Pythagorean theorem to find the length of the perpendicular segment OC:

OC² + AC² = OA²

OC² + 4² = 5²

OC² + 16 = 25

OC² = 25 − 16 = 9

∴ OC = 3 cm

The segment OD is a radius of the circle because it connects the centre O to a point D on the circumference. Therefore, the length of OD is equal to the radius OA, which is 5 cm.

The length of segment CD is the difference between the radius OD and the segment OC:

CD = OD − OC

= 5 cm − 3 cm

∴ CD = 2 cm

Thus, the length of CD is 2 cm.