Draw a line segment of lengths …

Given: A line segment of length 7.6 cm.
Required: To divide it in the ratio 5 : 8 and to measure the two parts.
 
Steps of construction :

1. From any ray AX, making an acute angle with AB.
2. Locate 13 (= 5 + 8) points A₁, A₂, A₃,..... and A₁₃ on AX such that
   AA₁ = A₁A₂ = A₂A₃ = A₃A₄ = A₄A₅ = A₅A₆ = A₆A₇ = A₇A₈ = A₈A₉ = A₉A₁₀ = A₁₀A₁₁ = A₁₁A₁₂ = A₁₂A₁₃
3. Join BA₁₃
4. Through the point A₅, draw a line parallel to A₁₃B intersecting AB at the point C.
   Then, AC : CB = 5 : 8
   On measurement, AC = 3.1 cm, CB = 4.5 cm.

Justification :
{tex}\because{/tex} A₅C || A₁₃B [ By Construction]
{tex}\because{/tex} {tex}\frac { A A _ { 5 } } { A _ { 5 } A _ { 13 } } = \frac { A C } { C B }{/tex} [By the Basic proportionality theorem]
But, {tex}\frac { A A _ { 5 } } { A _ { 5 } A _ { 13 } } = \frac { 5 } { 8 }{/tex} [ By Construction[
Therefore, {tex}\frac { A C } { C B } = \frac { 5 } { 8 }{/tex}
This shows that C divides AB in the ratio 5 : 8.