A medium question on counting shapes for P3–P4, 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
Inside a large triangle, 2 lines are drawn from the top corner to the base, and 1 line across, parallel to the base, as shown.
How many four-sided shapes are there in the picture?
Show the hints (2)
💡Hint 1: In every four-sided shape here, the top and bottom lie on across lines (the base counts), and the left and right on lines from the top corner.
💡Hint 2: Count the across lines (base included) and pick two for the top and bottom; then count the lines leaving the top and pick two for the left and right.
Worked solution
Have a go first, then read the solution step by step.
1.There are only 2 across lines (the line and the base): 1 way to pick the top and bottom.
2.4 lines leave the top (the two sides and 2 more); pick two: 3 + 2 + 1 = 6 ways.
3.Each top-and-bottom pick with each left-and-right pick makes one four-sided shape (the shaded one is one of them): 1 × 6 = 6 in all.
⚠️Common mistake: counting only the small pieces and missing the big shapes spanning several, or taking the triangles at the top for four-sided shapes.
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