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In fig. 3, 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 the side AD.
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Given: A circle touches all the four sides of a quadrilateral ABCD whose sides are AB=6cm, BC=9 cm and CD$=8$ cm in the given fig.
To do: To find AD$=?$
Solution: 

AB$=6\ $cm
BC$=9\ $cm
CD$=8$ cm AD$=?$
AB, BC, CD, AD, are tangents to the circle
And AP$=$AS, RD$=$DS
BP$=$BQ, AS$=$AP
CQ$=$CR
Also AB$=$AP$+$PB$=$ AP$+$PQ \ ……………….$( 1)$
BC$=$BQ$+$QC ……………………….. $( 2)$
CD$=$RC$+$DR  …………………………$( 3)$
AD$=$AS$+$DS  ………………………..$( 4)$
Adding$\ ( 1) ,\ ( 2) ,\ ( 3) ,\ ( 4)$
6$+$9$+$8$+$AD$=$AP$+$AS$+$BP$+$BQ$+$CQ$+$RC$+$RD$+$DS
23$+$AD$=2( AP) +2( BP) +2( RC) +2( RD)$
$23+AD=2( AB) +2( CD)$
$AD\ =5$ cm
 

Updated on: 10-Oct-2022

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