In triangle XYZ, the bisector of angle XYZ intersects overline XZ at E if XY/YZ=3/4 an XZ=42, find the greatest integer value of XY. Thus far, I have determined that XE=18 and ZE=24 by the angle bisector theorem, but I am unsure how to find XY.

Chaim Ferguson 2022-10-16 Answered
In triangle XYZ, the bisector of X Y Z intersects X Z ¯ at E if X Y Y Z = 3 4 an X Z = 42, find the greatest integer value of XY.
Thus far, I have determined that X E = 18 and Z E = 24 by the angle bisector theorem, but I am unsure how to find XY.
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Answers (1)

periasemdy
Answered 2022-10-17 Author has 15 answers

The bisector theorem gives X E E Z = X Y Y Z = 3 4 , hence X Z = 42 implies X E = 18 and E Z = 24.
By the triangle inequality, X Y + Y Z has to be greater than X Z = 42, hence X Y = 3 7 ( X Y + Y Z ) has to be greater than 18. On the other hand, Y Z Y X has to be smaller than X Z = 42, hence the length of X Y is at most 3 42 = 126
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