Answer:

To find the length of AD, we can use the property that a circle touching all four sides of a quadrilateral is called an inscribed circle or an incircle.

In a quadrilateral, the length of each side is the sum of two adjacent sides minus the other two sides. Applying this property to the quadrilateral ABCD, we have:

AD = AB + BC - CD

Substituting the given values, we get:

AD = 6 cm + 9 cm - 8 cm

AD = 7 cm

Therefore, the length of AD is 7 cm.

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Verified answer

Answer:

[tex]\boxed{\bf\:AD = 5 \: cm \: } \\ [/tex]

Step-by-step explanation:

Let assume that quadrilateral ABCD which circumscribing a circle touches AB, BC, CD, DA at P, Q, R, S respectively.

Now, A is external point and AP and AS are tangents to a circle.

We know, Length of tangents drawn from external point are equal.

[tex]\implies\sf \: AP = AS - - - (1) \\ [/tex]

Similarly,

[tex]\implies\sf \: BP = BQ - - - (2)\\ \implies\sf \: CR = CQ - - - (3)\\ \implies\sf \: DR = DS - - - (4) \\ [/tex]

On adding equation (1), (2), (3) and (4), we get

[tex]\sf \: AP + BP + CR + DR=AS + BQ + CQ + DS \\ [/tex]

can be regrouped as

[tex]\sf \: (AP + BP) + (CR + DR)=(AS + DS) + (BQ + CQ) \\ [/tex]

[tex]\sf \: AB + CD=AD + BC \\ [/tex]

On substituting the values of AB, BC, CD, we get

[tex]\sf\: 6 + 8 = 9 + AD \\ [/tex]

[tex]\sf\: 14 = 9 + AD \\ [/tex]

[tex]\sf\: 14 - 9 = AD \\ [/tex]

[tex]\implies\sf\:AD = 5 \: cm \\ [/tex]

Hence,

[tex]\implies\sf\:\boxed{\bf\:AD = 5 \: cm \: } \\ [/tex]