Draw a line segment $A B$ of length $7 c m$ . Using ruler and compasses, find a point P on AB such that $\frac{A P}{A B} = \frac{3}{5} .$

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Solution

Steps of Construction:
1) Draw a line segment $A B = 7 c m$
2) Draw a ray $A X$ , making an acute angle $∠ B A X$ .
3) Along $A X$ , mark $5$ points (greater of $3$ and $5$ ) $A_{I}, A_{2}, A_{3}, A_{4}$ and $A_{5}$ such that $A A_{I} = A A_{2} = A_{2} A_{3} = A_{3} A_{4} = A A_{5}$
4) Join $A_{5} B$ .
5) From $A_{3}$ , draw $A_{3} P$ parallel to $A_{5} B$ (draw an angle equal to $∠ A A_{5} B$ ), meeting $A B$ in $P$ .

Hence, P is a point on AB such that $\frac{A P}{A B} = \frac{3}{5} .$

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