Question 1109591: In a triangle ABC the lengths of all three sides are positive integers. The point M lies on the side BC so that AM is the internal bisector of angle BAC. Also, BM = 2 and MC = 3. What are the possible lengths of the sides of the triangle ABC. Thank you for helping I have been stuck on this question for a while now and I feel I'm missing something.
Answer by greenestamps(13203)   (Show Source):
You can put this solution on YOUR website!
We know side BC is 5, because BM is 2 and MC is 3.
Since AM bisects angle BAC, AB/AC = BM/MC = 2/3.
Probably that is what you were missing, because that makes the problem relatively easy.
You just need to find integers AB and AC for which AB/AC = 2/3 and for which AB, AC, and BC can form a triangle.
For example, if AB=2 and AC=3, then the three sides would be 2, 3, and 5; but those lengths do not make a triangle.
But if AB=4 and AC=6, then the three sides are 4, 5, and 6; and that DOES make a triangle.
So there is one answer; but there are more (two more). I leave it to you to find them.
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Added in response to reader's question....
In a triangle, an angle bisector divides the opposite side into two parts whose lengths are proportional to the lengths of the two sides that form the angle.
For example, if AB=5 and BC=7, the bisector of angle B divides side AC into two parts whose lengths are in the ratio 5:7.
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