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.,

ODBD

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.

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