Question

To divide a line segment PQ in a certain ratio, we draw a ray PM. Why don’t we draw it with an obtuse angle?

• A. We cannot measure an obtuse angle with the given line segment.
• B. Drawing an obtuse angle would make my constructions very congested.
• C. The diagram would be very large.
• D. The textbook says we should draw an acute angle.

Answer 1

The correct option is B

Drawing an obtuse angle would make my constructions very congested.

Suppose we draw $PM$ such that the angle $\angle QPM$ is an obtuse angle. Next we mark $(m+n)$ arcs on the ray PM such that $P{P}_1$ = $P_1{P}_2$ = $P_2{P}_3$ = … = $P_{m+n-1}$$P_{m+n}$.

For our convenience let us assume $m+n=5$ so we can explain things much easier. (In our case the required ratio could be 1:4 or 2:3 or 3:2 or 4:1 etc...)

Now in the $△$$QP{P}_5$,

If, $\angle$$QP{P}_5$ is obtuse then $\angle$$Q{P}_{5}P$ and $\angle$$PQ{P}_5$ would become very small (compared to the case where $\angle$ $QP{P}_5$ is acute) and it would be difficult to draw a line parallel to $Q{P}_5$ (Because small angles and hence small radii of arcs are involved in construction of parallel lines)

This will make drawing of parallel lines very congested compared to drawing parallel lines on line segment PN with acute angle $\angle QPN$.

You can see from the diagram that $P_4{Q}^{\prime }\parallel P_{5}Q$ is more closely spaced compared to $P_4^{\prime }{Q}^{\prime }\parallel P_5^{\prime }Q$.

So we prefer to draw an acute angle to make it more clear and spacious i.e. to avoid congestion.