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

Solution

The correct option is B Drawing an obtuse angle would make my constructions very congested. Suppose we draw PM such that the angle ∠QPM is an obtuse angle. Next we mark (m+n) arcs on the ray PM such that PP1=P1P2=P2P3=…=Pm+n−1Pm+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 △QPP5, If, ∠QPP5 is obtuse then ∠QP5P and ∠PQP5 would become very small (compared to the case where ∠ QPP5 is acute) and it would be difficult to draw a line parallel to QP5 (Because small angles and hence small radii of arcs are involved in construction of parallel lines) This will make the drawing of parallel lines very congested compared to drawing parallel lines on line segment PN with acute angle ∠QPN. You can see from the diagram that P4Q′∥P5Q is more closely spaced compared to P′4Q′∥P′5Q. So we prefer to draw an acute angle to make it more clear and spacious i.e. to avoid congestion.

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