Question

in figure, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB = 6 cm
BC = 9 cm and CD = 8 cm. Find the length of side AD.
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Answer 1

Answer:

Length of side AD is 5cm

Step-by-step explanation:

Refer the figure,

Here, ABCD is a quadrilateral.

Let us assume that the quadrilateral ABCD touches the circle at points P, Q, R and S.

Thus, AB, BC, CD and AD are tangents to the circle

Property of tangent states that "Tangent drawn from an external point to a circle are equal in length".

Thus, we get

AP = AS

Let AP = AS = x

Also, BP = BQ, CQ = CR and DR = DS

Since AP = x

Thus, BP = AB - AP = $6 - x$

Now, BQ = BP =  $6 - x$

Also, CQ = CB - BQ = $9 - (\,6 - x)\, = 3 + x$

Now, CR = CQ = $3 + x$

Also, DR = CD - CR = $8 - (\,3 + x)\, = 5 - x$

Now, DS = DR = $5 - x$

Finally,

AD = AS + DS

AD = x + 5 - x

AD = 5 cm

Thus, length of side AD is 5cm