The radii of two concentric circles are 13 cm and 8 cm. AB is a diameter of the bigger circle and BD is a tangent to the smaller circle touching it at D and intersecting the larger circle at P on producing. Find the length of AP.

The figure for the above given condition is as shown below,

Now we will join OD, we get

So as per the given criteria,

AB = 26cm, BO = AO = 13cm, OD = 8cm……….(i)

Now consider smaller circle,

In this BD is tangent to the smaller circle given and OD is the radius of the smaller circle,

And we know in a circle tangent is perpendicular to the radius, i.e.,

OD⊥BD

⇒ ∠BDO = 90°….(ii)

Now consider bigger circle,

In this P is a point in the semicircle with radius AB,

And we know in a circle, angle in a semicircle is always a right angle, i.e.,

∠APB = 90°….(iii)

Now we will consider ΔABP and ΔOBD,

∠APB = ∠BDO = 90° (from equation(ii) and (iii))

∠ABP = ∠DBO (common angle)

Hence by AA similarity,

ΔABP~ΔOBD

And we know sides of similar triangles are proportional, hence in these two triangles,

Now substituting values from equation (i), we get

⇒ AP = 2 × 8

⇒ AP = 16cm

Hence the length of AP is 16cm.