How to identify two points P and Q on a line segment AB such that AP:PQ = 1:2 and PQ:QB = 4:5?
Find the ratios in which the points P and Q divide line segment AB and then divide the line segment AB in the ratios obtained.
Given AP:PQ = 1:2 and PQ:QB = 4:5.
APPQ=12⇒APPQ+1=12+1⇒AP+PQPQ=32⇒AQPQ=32−−−−−−−−−−−−−(1)QBPQ=54−−−−−−−−−−−−−−−(2)
Dividing equation (1) by (2)
⇒AQQB=65−−−−−−−−−−−−(3)
Hence Q divides AB in the ratio of 6:5.
Adding equations (1) and (2),
⇒AQPQ+QBPQ=32+54⇒ABPQ=114−−−−−−(4)
Dividing APPQ=12 with equation (4), we get
APAB=211
Hence P divides line segment AB in the ratio of 2:11.
Now proceed further by dividing line segment AB by the two ratios respectively and thereby identifying the points.