Problem 16 · 2014 Math Kangaroo
Stretch
Geometry & Measurement
substitution
In triangle ABC (see sketch) AD is the angle bisector of the angle at A, and BH is the height from side AC. The obtuse angle between BH and AD is four times the size of angle \(\angle DAB\). How big is the angle \(\angle CAB\)?
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Answer: C — \(60°\)
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Hint 1 of 2
Let angle DAB = α, so angle CAB = 2α; the height BH is perpendicular to AC.
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Hint 2 of 2
Find the angle between AD and the perpendicular BH in terms of α, then set its obtuse value equal to 4α.
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Approach: express the angle between the bisector and the height
- Let ∠DAB = α, so ∠CAB = 2α and AD makes angle α with AC.
- BH is perpendicular to AC, so the acute angle between AD and BH is 90° − α, and the obtuse one is 90° + α.
- Set 90° + α = 4α, giving 3α = 90°, so α = 30°.
- Then ∠CAB = 2α = 60°.
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