Maths olympiad question · P6★★★★ Expert
Perimeter and area: olympiad question 11
An expert question on perimeter and area for P6, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P4 to P6 papers; harder ones work out areas from the ratios between triangles' areas.
The question
In the figure, triangle ABC has an area of 2652 cm².
D is on BC with BD :
DC = 2 :
1;
E is on CA with CE :
EA = 1 :
5.
AD and BE cross at O.
What is the area of quadrilateral ODCE (shaded), in cm²? (Not to scale.)
Show the hints (3)
- Hint 1: Join OC, splitting the big triangle into AOB, BOC and COA. AOB and AOC share the side AO: their areas are in the ratio of what?
- Hint 2: AOB : AOC = BD : DC = 2 : 1; likewise AOB : BOC = AE : EC = 5 : 1. Chain the two ratios.
- Hint 3: AOB : AOC : BOC = 10 : 5 : 2. Then cut BOC and COA once more at D and E.
Worked solution
Have a go first, then read the solution step by step.
- 1.Join OC. Triangles AOB and AOC share AO, and their heights to it are as BD : DC, so AOB : AOC = 2 : 1.
- 2.Likewise AOB and BOC share BO: AOB : BOC = AE : EC = 5 : 1.
- 3.Chained: AOB : AOC : BOC = 10 : 5 : 2, 17 parts of 2652 ÷ 17 = 156 cm²: AOB = 1560, AOC = 780, BOC = 312 cm².
- 4.OD cuts BOC in the ratio BD : DC, so COD = 312 × 1/3 = 104 cm².
- 5.OE cuts COA in the ratio CE : EA, so COE = 780 × 1/6 = 130 cm².
- 6.ODCE = 104 + 130 = 234 cm².
- ⚠️Common mistake: assuming O cuts AD in the ratio 2 : 1 or 1 : 5 and slicing the big triangle with it. How O divides a line comes from the area ratios, not straight from the sides.
Answer: 234 cm²
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