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?

Dangelo Rosario 2022-10-09 Answered
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?
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Answers (1)

Johnny Parrish
Answered 2022-10-10 Author has 12 answers
Step 1
Once you'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 2
Solve 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.
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