AB and CD are two common tangents of two circles which touch each other at C. If D lies on AB and CD=5 cm, then what is the length of AB.

DC and AB are tangents given to both circle

Point D is on AB which means DA and DB are also tangents to both circle

Now from point D, we have two tangents to bigger circle which are DA and DC

⇒ DA = DC …tangents from a point to a circle are equal

⇒ DA = 5 cm …DC is 5 cm given…(i)

Also from point D, we have two tangents to smaller circle which are DB and DC

⇒ DB = DC …tangents from a point to a circle are equal

⇒ DB = 5 cm …DC is 5 cm given…(ii)

Now as point D is on AB from the figure we can say that DA + DB = AB

⇒ AB = DA + DB

Using (i) and (ii)

⇒ AB = 5 + 5

⇒ AB = 10 cm

Hence the length of tangent AB is 10 cm