Problem 30 · 2009 Math Kangaroo
Stretch
Geometry & Measurement
symmetry
In triangle ABC the interior angle B equals 20° and C equals 40°. The length of the angle bisector through A is 2. What is the difference of the side lengths BC and AB?
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Answer: C — 2
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Hint 1 of 2
With B=20 and C=40, angle A is 120, so its bisector makes two 60 degree halves.
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Hint 2 of 2
Set the sides by the sine rule with bisector length 2; the difference BC - AB is strikingly clean.
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Approach: use the angle-bisector length
- Angle A = 180 - 20 - 40 = 120, bisected into two 60 degree parts; by the sine rule the sides are proportional to sin20, sin40, sin120.
- Scaling so the bisector from A has length 2 and computing the sides, BC - AB works out to exactly 2.
- The difference of side lengths BC and AB is 2.
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