How many triangles can be found in a figure consisting of a triangle with two lines drawn from the apex to the base, dividing it into three parts?
Correct answer: (B) 6.
A triangle divided from its apex to the base by two internal lines contains six triangles in all: three small triangles, two medium triangles, and the outer triangle.
Explanation
The figure should be counted by grouping triangles by size, then adding non-overlapping categories. The two lines from the apex to the base split the original triangle into three small triangles. Combining two adjacent small triangles gives two medium triangles: one on the left pair and one on the right pair. Finally, the original boundary still forms one large outer triangle. NCERT's counting principle supports this method: when cases are separate, their counts are added. Here the separate cases are small, medium, and large triangles, so the total is 3 + 2 + 1 = 6.
Why the other options are wrong
- (A) Three counts only the smallest triangles and ignores the two medium triangles and the large outer triangle.
- (C) Four adds only one more triangle to the three small ones, but there are three additional larger triangles.
- (D) Nine overcounts because the only valid triangles are the three small, two medium, and one large triangle formed by the given lines.
Concept
This tests visual counting in Reasoning and Mental Ability, using the same addition principle applied in elementary counting problems. It recurs in RAS because the answer depends on a systematic classification of cases, not on guessing from the diagram.