In triangle ABC, A B = 84 , B C = 112, and A C = 98. Angle B is bisected by line segment BE, with point E on AC. Angles ABE and CBE are similarly bisected by line segments BD and BF, respectively. What is the length of FC?

Question

In triangle ABC,   A  B  =  84  ,  B  C  =  112, and   A  C  =  98. Angle B is bisected by line segment BE, with point E on AC. Angles ABE and CBE are similarly bisected by line segments BD and BF, respectively. What is the length of FC?

Johnny Parrish · Accepted Answer

Step 1Once you&#039;ve used the angle bisector theorem you know that the center bisector breaks up 98 into 42 and 56. Now drop a height to the 98 side. You know that it has to be closer to the 84 side than that center bisector. Why? Now break up the base into   42  −  x and   56  +  x according to where you put the height. Using the Pythagorean theorem, we know       84    2    −  (  42  −  x      )    2    =      112    2    −  (  56  +  x      )    2  .Step 2Solve and you get   x  =  21. This means that the altitude (height)of the triangle formed by the 84 side, the 42 side, and the center bisector is also a median. Therefore, that triangle is isosceles and the middle bisector has length 84 and the result follows from another application of the angle bisector theorem.

In triangle ABC, AB=84, BC=112, and AC=98. Angle B is bisected by line segment BE, with point E on AC. Angles ABE and CBE are similarly bisected by line segments BD and BF, respectively. What is the length of FC?

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