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.

Question

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.

periasemdy · Accepted Answer

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

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.

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