RAS question
In the given figure, a pentagon with all its diagonals drawn, how many straight line segments are there?
Correct answer: (C) 10.
A pentagon with all its diagonals drawn has 10 straight line segments: 5 sides and 5 diagonals.
Explanation
The figure asks for all straight line segments joining the pentagon's vertices. A pentagon has 5 sides. To count every vertex-to-vertex segment, choose any 2 of its 5 vertices; NCERT's combinations rule applies because order does not matter in a selection. This gives C(5,2) = 10 total vertex-to-vertex segments. Of these, 5 are already the sides of the pentagon. The remaining 10 - 5 = 5 are diagonals. Therefore, the total number of straight line segments in the figure is 5 sides + 5 diagonals = 10.
Why the other options are wrong
- (A) 8 undercounts the figure because, after the 5 sides, a pentagon still has 5 diagonals, not just 3.
- (B) 5 counts only one category, either the sides alone or the diagonals alone, while the figure contains both.
- (D) 15 overcounts because C(5,2) = 10 already gives all vertex-to-vertex line segments; the 5 sides should not be added again to that total.
Concept
This tests basic combinatorial counting in geometry: using combinations to count line segments determined by pairs of vertices. It recurs in RAS reasoning because figure-counting questions often reduce to choosing pairs without regard to order.