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Question
ABCD is a trapezium in which AB || DC and P and Q are points on AD and BC, respectively such that PQ || DC. If PD = 18 cm, BQ = 35 cm and QC = 15 cm, find AD.
Solution
Given, a trapezium ABCD in which AB || DC.
P and Q are points on AD and BC, respectively such that PQ || DC.
Thus, AB || PQ || DC.
Join BD.
In ΔABD,
PO || AB ...[∵ PQ || AB]
By basic proportionality theorem,
`("DP")/("AP") = ("DO")/("OB")` ...(i)
In ΔBDC,
OQ || DC ...[∵ PQ || DC]
By basic proportionality theorem,
`("BQ")/("QC") = ("OB")/("OD")`
⇒ `("QC")/("BQ") = ("OD")/("OB")` ...(ii)
From equation (i) and (ii),
`("DP")/("AP") = ("QC")/("BQ")`
⇒ `18/("AP") = 15/35`
⇒ AP = `(18 xx 35)/15` = 42
∴ AD = AP + DP
= 42 + 18
= 60 cm
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