Question

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

• A. Draw a ray at an acute angle to AB and identify the points P and Q on AB with ratio 1:2 and 4:5 respectively.
• 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

Answer: (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 $\frac{AP}{PQ}=\frac{1}{2}$ with equation (4), we get
$\frac{AP}{AB}=\frac{2}{11}$
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.