Problem 26 · 2018 Math Kangaroo
Stretch
Geometry & Measurement
area-fractionsymmetry
The points N, M and L lie on the sides of an equilateral triangle ABC so that NM ⊥ BC, ML ⊥ AB and LN ⊥ AC. The area of triangle ABC is 36 cm². What is the area of triangle LMN?
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Answer: B — 12 cm²
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Hint 1 of 2
The three right-angle conditions place L, M, N symmetrically on the three sides.
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Hint 2 of 2
Inner triangle LMN turns out to be a fixed fraction of triangle ABC.
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Approach: exploit the symmetric perpendicular-foot construction
- The conditions NM ⊥ BC, ML ⊥ AB and LN ⊥ AC place L, M, N symmetrically, so triangle LMN is equilateral and centred in ABC.
- For this construction the inner triangle has exactly one third of the area of ABC.
- So its area is 36 : 3 = 12 cm².
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