Question

The parallelogram ABCD shows the points P and Q dividing each of the lines AD and DC in the ratio 1:4. What is the ratio in which R divides DB? What is the ratio in which R divides PQ?

Answer 1

see explanation.

Explanation:

Let `|ABC|` denote area of `DeltaABC`
let `|ABCD|=10x, => |ABC|=|ADC|=|ABD|=|CDB|=(10x)/2=5x`
given `DQ:QC=1:4, => |ADQ|:|AQC|=|BDQ|:|BQC|=1:4`,
`=> |ADQ|=x, |AQC|=4x, and |BDQ|=x, |BQC|=4x`
given `AP:PD=1:4, => |AQP|:|PQD|=|ABP|:|PBD|=1:4`,
`=> |ABP|=x, |PBD|=4x`
and `|AQP|=1/5x, |PQD|=4/5x`
`|PDQB|=|PBD|+|BDQ|=4x+x=5x`
`=> |PQB|=|PDQB|-|PDQ|=5x-4/5x=21/5x`
`=> DR:RB=|PDQ|:|PQB|=4/5x:21/5x=4:21`

`PR:RQ=|BPD|:|BDQ|=4x:x=4:1`