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
Triangle ABC has 1 more point on side AB, 1 on side BC and 4 on side CA.
With the three corners, there are 9 points in the picture, as shown.
How many different triangles have three of these 9 points as their corners?
Show the hints (3)
💡Hint 1: How many ways are there to pick three of the 9 points? Which picks don't make a triangle?
💡Hint 2: Three points on the same side make no triangle. How many points does each side have, its two corners included?
💡Hint 3: AB has 3 points: 1 way. Work out the other two sides the same way and take them all off the total.
Worked solution
Have a go first, then read the solution step by step.
1.Three of 9 points (the shaded triangle is one pick): 9 × 8 × 7 ÷ 6 = 84 ways.
2.Three on one side make no triangle, so take those off: AB has 3 points: 1 way; BC has 3 points: 1 way; CA has 6 points: 6 × 5 × 4 ÷ 6 = 20 ways. 22 in all.
3.Triangles: 84 − 22 = 62.
⚠️Common mistake: forgetting the corners are on the sides. A, for one, is on both AB and CA, so A and any two points of AB lie on one line.
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