Question

Draw a line segment AB of length 10 cm and divide it internally in the ratio of 2:5 Justify the division of line segment AB.

Answer 1

Steps of construction:

1. Draw a line segment AB of length 10 cm.
2. Draw a line AX making an acute angle with AB.
3. Mark 7 points A₁, A₂, A₃, ...... A₇ along AX such that AA₁ = A₁A₂ = A₁A₃ = ...... = A₆A₇.
4. Join point B with point A₇.
5. Through the point A₂, draw a line parallel to A₇B by making an angle equal to ∠AA₇B at A₂. This line meets AB at a point P.

As a result, point P is necessary to divide line AB internally in the ratio of 2 : 5.

Justification: In ΔAA₇B, A₂P || A₇B.

As a result of the basic proportionality theorem,

${\displaystyle \frac{A{A}_2}{A_2{A}_7}=\frac{AP}{PB}}$  .......(i)

By the above construction,

${\displaystyle \frac{A{A}_2}{A_2{A}_7}=\frac{2}{5}}$  ......(ii)

By equations (i) and (ii),

${\displaystyle \frac{AP}{PB}=\frac{2}{5}}$

Thus, AP : PB = 2 : 5

As a result, P splits the line AB in the ratio 2 : 5.