Problem 25 · 2015 Math Kangaroo
Stretch
Geometry & Measurement
pythagorean-triplearea
In a right-angled triangle the angle bisector of an acute angle splits the opposite side into segments of length 1 and 2 respectively. How long is this angle bisector?
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Answer: C — \(\sqrt{4}\)
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Hint 1 of 3
By the angle-bisector theorem, the two segments (1 and 2) are in the ratio of the two sides meeting at that acute vertex.
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Hint 2 of 3
Put the right angle at the origin and the foot of the bisector on a leg, then read off the bisector's length as a distance.
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Hint 3 of 3
The ratio of sides is 2 : 1, which fixes the triangle; the bisector then comes out a clean length.
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Approach: use the angle-bisector ratio, then place coordinates
- The bisector of the acute angle at B meets the opposite leg AC (length 1 + 2 = 3) at D, with AD : DC = BA : BC.
- Put the right angle at C = (0,0), A = (3,0), B = (0,√3); then BA = 2√3 and BC = √3, a 2 : 1 ratio, so D is at distance 1 from C: D = (1,0).
- The bisector is BD = distance from (0,√3) to (1,0) = √(1 + 3) = 2.
- So the bisector has length √4 (= 2).
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