Assertion (A): In the given figure, chord AB = 8 cm, diameter CD = 20 cm, then length of OP = 10 cm.

Reason (R): OP = $\sqrt{OA^2 - AP^2}$

and CP = OC + OP

1. A is true, but R is false.

2. A is false, but R is true.

3. Both A and R are true, and R is the correct reason for A.

4. Both A and R are true, and R is the incorrect reason for A.

Given:

Length of the chord AB = 8 cm.

Diameter of the circle CD = 20 cm.

As we know that the radius of a circle is exactly half its diameter.

Radius of the circle, r = $\dfrac{20}{2}$ = 10 cm.

Construction: Join OA.

OP ⊥ AB.

Since, perpendicular drawn from the center of a circle to a chord bisects it.

∴ OP bisects AB

⇒ AP = $\dfrac{1}{2}$ x AB = $\dfrac{1}{2}$ x 8 = 4 cm

⇒ OA = 10 cm

In Δ OAP, ∠P = 90°

Using Pythagoras theorem,

∴ OA² = OP² + AP²

⇒ OP² = OA² - AP²

$\Rightarrow OP = \sqrt{OA^2 - AP^2}\\[1em] \Rightarrow OP = \sqrt{10^2 - 4^2}\\[1em] \Rightarrow OP = \sqrt{100 - 16}\\[1em] \Rightarrow OP = \sqrt{84}\\[1em] \Rightarrow OP = 2\sqrt{21} \text{ cm}$

∴ Assertion (A) is false.

From figure, CP = CO + OP = 10 + 2 $\sqrt{21}$ cm

∴ Reason (R) is true.

∴ A is false, but R is true.

Hence, option 2 is the correct option.