Question

Determine a point which divides a line segment of length $12cm$ internally in the ratio $2:3$. Also, justify you construction.

Answer 1

Steps of Construction

1. Draw a line segment $AB$ of $12cm$.

2. Through the points $A$ and $B$ draw two parallel lines on the opposite side of $AB$.

3. Cut two equal parts on $AX$ and three equal parts on $BY$ such that $A{A}_1=A_1{A}_2$ and $B{B}_1=B_1{B}_2=B_2{B}_3$.

4. Join $A_2{B}_3$ which intersects $AB$ at $P$.

Therefore by construction $\frac{AP}{BP}=\frac{2}{3}$

Justification :

1. In $△A{A}_{2}P$ and $△B{B}_{3}P$, we have

$\angle AP{A}_2=\angle BP{B}_3$ (because they are vertically opposite angles)

$\angle A_{2}AP=\angle B_{3}BP$ (Alternate interior angles are equal)

$△A{A}_{2}P$ and $△B{B}_{3}P$ are similar to each other (Angle Angle similarity)

Therefore $\frac{AP}{BP}=\frac{A{A}_2}{B{B}_3}=\frac{2}{3}$