Question

The intercepts made by 3 parallel lines on a transversal line (\[{{l}_{1}}\]) are in the ratio 1: 1. A second transverse line (\[{{l}_{2}}\]) making an angle of $30{}^\circ $ with ${{l}_{1}}$ is drawn. The corresponding intercepts on \[{{l}_{2}}\]​are in the ratio:
A. 1:1
B. 2:1
C. 1:2
D. 1:3

Answer 1

Hint: For solving this question we have to extend the transversals and we have to join the two transversals below the transversals to form a triangle. Then by the triangle proportionality theorem we have to find the ratio of corresponding intercepts on \[{{l}_{2}}\].

Complete step-by-step solution:
For the given problem we are given to find the ratio of the intercepts on \[{{l}_{2}}\]. For that we should know the triangle proportionality theorem which states that if a line is parallel to one side of a triangle and it intersects the other two sides, then it divides those sides proportionally.
We have to extend the transversals \[{{l}_{1}}\] and \[{{l}_{2}}\], then make them join at a point A. By joining the other two sides we can form a triangle.
Let us draw a rough diagram for the given question.

Therefore, from the graph we can say that
\[\Rightarrow \dfrac{AB}{BC}=\dfrac{PQ}{QR}\]
Let us consider the equation as equation (1).
\[\dfrac{AB}{BC}=\dfrac{PQ}{QR}...........\left( 1 \right)\]
From the question it is given that
\[\dfrac{AB}{BC}=\dfrac{1}{1}\]
Therefore from the equation (1) we can say that
\[\dfrac{PQ}{QR}=\dfrac{1}{1}\]
Hence, the correct option is A.

Note: We can solve this problem by another concept i.e. if three parallel lines are intercepted by 2 transversals, in such a way that the two intercepts on one traversal are equal, then the intercepts formed by the other transversal are also equal. Such types of questions need more practice. We should remember all concepts related to lines so that if this question comes in twisted form so we can tackle that easily. Solving this question with a lot of focus will reward us full marks. Thus, it is important to solve it with focus to get the right answer.