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.

Answer 1

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