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.
• 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.
• 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.
• D. None of the above

Answer 1

The correct option is 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.

Given $AP:PQ=1:2$ and $PQ:QB=4:5$.
$\frac{AP}{PQ}=\frac{1}{2}\Rightarrow \frac{AP}{PQ}+1=\frac{1}{2}+1\Rightarrow \frac{AP+PQ}{PQ}=\frac{3}{2}\Rightarrow \frac{AQ}{PQ}=\frac{3}{2}...\left(1\right)\frac{QB}{PQ}=\frac{5}{4}...\left(2\right)$
Dividing equation (1) by (2)
$\Rightarrow \frac{AQ}{QB}=\frac{6}{5}...\left(3\right)$
Hence Q divides AB in the ratio of 6:5.
Adding equations (1) and (2),
$\Rightarrow \frac{AQ}{PQ}+\frac{QB}{PQ}=\frac{3}{2}+\frac{5}{4}\Rightarrow \frac{AB}{PQ}=\frac{11}{4}...\left(4\right)$
Dividing $\frac{AP}{PQ}=\frac{1}{2}$ by equation (4), we get
$\frac{AP}{AB}=\frac{2}{11}$
$\Rightarrow$ 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.