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, $AB=84,BC=112$, 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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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.

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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