Question

A line segment CD has two points R and S such that CR:RD = 2:3 and CS:SD = 5:6.

Sort the following segments in ascending order of their lengths.
$CR$, $CS$, $RD$, $SD$

Answer 1

Given $CR:RD=2:3$ and $CS:SD=5:6$

Let us try to find all the line segments with respect to CD.

$\frac{CR}{RD}=\frac{2}{3}---\left(1\right)\Rightarrow \frac{CR}{RD}+1=\frac{2}{3}+1\Rightarrow \frac{CR+RD}{RD}=\frac{5}{3}\Rightarrow \frac{CD}{RD}=\frac{5}{3}\Rightarrow RD=\frac{3}{5}CD---\left(2\right)$

From equation (1)
$\Rightarrow CR=\frac{2}{3}RD=\frac{2}{3}\times \frac{3}{5}RD$
$=\frac{2}{5}CD---\left(3\right)$

$\Rightarrow \frac{CS}{SD}=\frac{5}{6}---\left(4\right)\Rightarrow \frac{CS}{SD}+1=\frac{5}{6}+1\Rightarrow \frac{CS+SD}{SD}=\frac{11}{6}\Rightarrow \frac{CD}{SD}=\frac{11}{6}\Rightarrow SD=\frac{6}{11}CD---\left(5\right)$

From equation (4)
$\Rightarrow CS=\frac{5}{6}SD=\frac{5}{6}\times \frac{6}{11}CD$
$=\frac{5}{11}CD---\left(6\right)$

From equations (1), (2), (3) and (4), converting all the segments to common denominator.
$RD=\frac{33}{55}CDCR=\frac{22}{55}CDSD=\frac{30}{55}CDCS=\frac{25}{55}CD$

Hence,
$CR<CS<SD<RD$