In the given figure, PB is a tangent to a circle with centre O at B. AB is a chord of length 24 cm at a distance of 5 cm from the centre. If the length of the tangent is 20 cm, find the length of OP.

Join OB as shown in figure below:

OM = 5 cm

OM ⊥ AB and M is mid-point of AB,

MB = $\dfrac{1}{2}AB = \dfrac{1}{2} \times 24 =$ 12 cm.

In right-angled triangle △OMB,

$OB^2 = OM^2 + MB^2 \\[1em] \Rightarrow OB^2 = 5^2 + 12^2 \\[1em] \Rightarrow OB^2 = 25 + 144 \\[1em] \Rightarrow OB^2 = 169 \\[1em] \Rightarrow OB = \sqrt{169} \text{ cm} \\[1em] OB = 13 \text{ cm.}$

As BP is tangent to circle at B, OB ⊥ BP.

In right-angled triangle △OBP,

$OP^2 = OB^2 + BP^2 \\[1em] \Rightarrow OP^2 = 13^2 + 20^2 \\[1em] \Rightarrow OP^2 = 169 + 400 \\[1em] \Rightarrow OP^2 = 569 \\[1em] \Rightarrow OP = \sqrt{569} \text{ cm}$

Hence, the length of OP = $\sqrt{569}$ cm.