How many triangles are there in the given figure?
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Answer: Option C — 12
Detailed Solution
Label the rectangle ABCD (A bottom-left, B bottom-right, C top-right, D top-left). The diagonals meet at the centre O, and the vertical middle line MN (M on AB, N on DC) also passes through O.
Smallest triangles: AOD, BOC, AMO, MBO, DNO, NOC = 6. Two-piece triangles: AOB and DOC = 2.
Half-rectangles cut off by a diagonal: ABC, ABD, ACD, BCD = 4. Total = 6 + 2 + 4 = 12.
✓ Check
The middle line only splits AOB and DOC, adding 4 small pieces; it creates no triangle with A–D or B–C sides.⚠ Common trap
(A) 8 — misses the half-rectangles ABC, ABD, ACD and BCD made by the full diagonals.Quick tip
Count by size (single regions, then pairs, then larger) or by fixing one vertex at a time
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Counting of Figures — Preparation
This question tests your understanding of Counting of Figures, a key topic in the Logical Reasoning section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.
Counting of figures questions ask you to count how many triangles, squares, rectangles or straight lines a complex figure contains. They look simple but are easy to get wrong because shapes overlap and combine. The most reliable approach is systematic counting. Label every point where lines meet, then list the shapes by size: first the smallest single shapes, then shapes made of two parts, then three, and so on. Formulas save a lot of time for standard figures. A triangle with n lines drawn from its top vertex to the base contains n(n + 1)/2 triangles if the n lines include the two sides. A grid of m by n squares contains m(m + 1)n(n + 1)/4 rectangles. Never try to count by eye alone, because it is easy to miss shapes formed by combining smaller ones. Practise the counting of figures questions below. Each solution shows a labelled figure and the count step by step.