A hard question on counting shapes for P4–P5, with a step-by-step solution, diagrams and the answer. Every year of the Hua Xia Cup, P1 to P6, has shape-counting questions. The hard part is counting each shape once and missing none.
The question
In triangle ABC, 2 lines are drawn from corner B to the opposite side AC, and 1 line from corner C to the opposite side AB, as shown.
How many triangles are there in the picture?
Show the hints (3)
💡Hint 1: Each triangle's three sides lie on three different lines. Which lines pass through B? Which through C? Can three lines all through B make a triangle?
💡Hint 2: So the triangles come in two kinds: two lines through B and one through C, or one through B and two through C. Careful: BC passes through both B and C.
💡Hint 3: 4 lines pass through B (BA, BC and the 2 drawn from B); 2 pass through C but not B. Triangles using BC get counted in both kinds.
Worked solution
Have a go first, then read the solution step by step.
1.Through B: BA, BC and the 2 drawn from B, 4 lines. Through C: CA, CB and the 1 drawn from C, 3 lines. BC is in both.
2.A triangle's three sides lie on three different lines, not all through one point. First kind: two through B and one through C (not BC), like the shaded one: two of 4 is 3 + 2 + 1 = 6 ways, times 2 lines: 6 × 2 = 12.
3.Second kind: one through B (not BC) and two through C, like the shaded one: two of 3 is 2 + 1 = 3 ways, times 3 lines: 3 × 3 = 9.
4.Triangles on BC (BC, another line through B, another through C, like the shaded one) were counted in both kinds: 3 × 2 = 6, so take them off once.
5.12 + 9 − 6 = 15 triangles in all.
⚠️Common mistake: counting only the smallest triangles, or splitting into two kinds and forgetting that the triangles on BC were counted twice.
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