ABCD is a trapezium in which AB`abs()`DC and P,Q are points on AD and BC respectively, such that PQ`abs()`DC, if PD=18 cm, BQ=35 cm and QC=15 cm. Find AD.

Explanation

Given, a trapezium ABCD in which AB`abs()`DC. Pa nd Q are points on AD and BC, respectively such that `PQabs()DC`. Thus, `ABabs()PQabs()DC`

Join BD
In `DeltaABD,POabs()AB` `[becausePQabs()AB]`
By basic proportionality theorem, `(DP)/(AP)=(DO)/(OB)` ....(i)
In `DeltaBDC, OQabs()DC[becausePQabs()DC]`
By basic proportionality theorem,
`(BQ)/(QC)=(QB)/(OD)`
`rArr` (QC)/(BQ)=(OD)/(OB)` ......(ii)
From Eqs. (i) and (ii). (DP)/(AP)=(QC)/(BQ)`
`rArr 18/(AP)=15/35`
`rArrAP=(18xx35)/15=42`
`therefore AD+AP+DP=42+18=60 cm`