Find the number of triangles in the given figure.
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Find the number of triangles in the given figure.
Answer: D. 27 — Concept: In a figure made of straight lines, a triangle is any set of three of those lines that cross in three distinct points, with all three of those points…
- A.
21
- B.
23
- C.
25
- D.
27
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Show answer & explanation
Correct answer: D
Concept: In a figure made of straight lines, a triangle is any set of three of those lines that cross in three distinct points, with all three of those points lying on drawn segments. A count is reliable only when the triangles are grouped by size — by how many of the smallest triangular cells each one covers — because every triangle then belongs to exactly one group, so none is skipped and none is counted twice.
Applying it here: The drawing is a six-pointed star: two large overlapping triangles, one point-up and one point-down, whose overlap is a regular hexagon. Three internal straight lines are drawn. Each starts at the midpoint of one of the three upper hexagon sides, passes through the centre, and leaves through the opposite star point — so each internal line is a median of a large triangle: two medians on the point-up triangle and one median on the point-down triangle.
Taking the smallest triangle as 1 unit, every triangle falls into one of these size groups:
Size | What these triangles are | How many |
|---|---|---|
1 unit | Halves of the three star points that an internal line passes through | 6 |
2 units | The six star points themselves | 6 |
3 units | Two internal lines together with one side of a large triangle | 6 |
6 units | The point-up triangle’s base together with its two medians | 1 |
9 units | Half of a large triangle, cut off by a drawn median (4 from the point-up triangle, 2 from the point-down triangle) | 6 |
18 units | The two large overlapping triangles | 2 |
Adding the groups: 6 + 6 + 6 + 1 + 6 + 2 = 27 triangles.
Cross-check: The drawing is built from exactly 9 straight lines. Testing all 84 possible triples of those lines, and keeping only the triples that meet in three distinct points with all three points lying on drawn segments, again leaves 27 triangles. The triple made of the three internal lines is excluded by that test, because those lines are concurrent and their crossings coincide at the centre. The result agrees with the size-by-size tally.