In the figure, a circle touches all the four sides of a quadrilateral ABCD with AB = 6 cm, BC = 7 cm, and CD = 4 cm. Find AD.
In the figure, a circle touches all the four sides of a quadrilateral ABCD with AB = 6 cm, BC = 7 cm, and CD = 4 cm. Find AD.

Given,

A circle touches the sides AB, BC, CD and DA of a quadrilateral ABCD at points P, Q, R and S respectively.

AB=6cm,

BC=7cm,

CD=4cm

Let AD=X

AsAP  and AS are the tangents.
AP=AS

Similarly,

BP=BQ

CQ=CR

and DR=DS

So, In ABCD

AB+CD=AD+BC  (This is the property of a cyclic quadrilateral)

(6+4=7+X)

\Rightarrow 10=7+X

\Rightarrow X=3

Therefore, AD=3cm.