Question

# How do we identify two points P and Q on a line segment AB such that AP : PQ = 1 : 2 and PQ : QB = 4 : 5?

A
The point P is mid point of AB and Q is mid point of line segment PB. Identify the points P and Q by dividing AB accordingly.
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B
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.
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C
The points P and Q are points of trisection of the line segment AB. Identify the points P and Q by dividing AB into three equal parts.
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D
None of the above
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Solution

## The correct option is A 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 by equation (4), we get APAB=211 ⇒ P divides line segment AB in the ratio of 2 : 9. Now proceed further by dividing line segment AB by the two ratios respectively and thereby identifying the points.

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