Construction 11.1 : To divide a line segment in a given ratio.
Let us divide a line segment B into 3:2 ratio.
Steps of construction:
Draw line AB
Draw any ray AX, making an acute angle (angle less than 90°) with AB.
Mark 5 (= 3 + 2) points 𝐴_1, 𝐴_2, 𝐴_3, 𝐴_4 and 𝐴_5 on AX so that 〖𝐴𝐴〗_1=〖𝐴_1 𝐴〗_2=〖𝐴_2 𝐴〗_3=〖𝐴_3 𝐴〗_4=〖𝐴_4 𝐴〗_5 by drawing equal arcs
Since we want the ratio 3 : 2, Through point 𝐴_3 (m = 3), we draw a line parallel to 𝐴_5 𝐵 (by making an angle equal to ∠AA5B at A3) intersecting AB at the point C.
Thus, AC : CB = 3 : 2.
In Δ AA5B
Since 𝐴_3 𝐶 is parallel to 𝐴_5 𝐵, therefore,
〖𝐴𝐴〗_3/(𝐴_3 𝐴_5 )=𝐴𝐶/𝐶𝐵
By construction, 〖𝐴𝐴〗_3/(𝐴_3 𝐴_5 )=3/2.
This shows that C divides AB in the ratio 3 : 2.