Steps of Construction
1. Draw a line segement AB=6cm.
2. Draw any ray making an acute `angleBAX` with AB.
3. Along AX, mark off ( larger among the ratios ) 5 points `A_(1),A_(2),A_(3),A_(4),A_(5)` such that `A A_(1)=A_(1)A_(2)=A_(2)A_(3)=A_(3)A_(4)=A_(4)A_(5)`.
4. Join the point `A_(2)(5-3)` with end point B.
5. Draw a line parallel to `A_(2)B` from `A_(5)` ( larger among the ratios ) which meets AB produced at P.
Point P, so obtained is the required point such that AP:BO=5:3.
Justification: In`DeltaA A_(5)P`,
Since `A_(2)B||A_(5)P` ( construction )
`:. (AP)/(BP)=(A A_(5))/(A_(2)A_(5))` ( By B.P. theorem )
`implies (AP)/(BP)=(5)/(3)` (construction )