Triangle angle bisector problem: Finding area (triangle ABC)=?

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2022-08-19

Triangle angle bisector problem: Finding area

Area A B C=?
How to apply this theorem?

Answer & Explanation

Douglas Woodard

Douglas Woodard

Beginner2022-08-20Added 8 answers

Step 1

Area A B C = ?
Defining x,y,z as shown in the figure above, we can write the following equation from the angle bisector theorem.
6 / x = ( 5 + y ) / ( 9 + z )
5 / y = ( 6 + x ) / ( 9 + z )
9 / z = ( 6 + x ) / ( 5 + y )
therefore, 6 ( 9 + z ) = x ( 5 + y )
5 ( 9 + z ) = y ( 6 + x )
9 ( 5 + y ) = z ( 6 + x )
Step 2
These equation have only one valid solution, namely, x = 9 , y = 5 , z = 6.
Hence the sides of the triangle, are 15,10,15
the semi-perimeter is s = 1 2 ( 15 + 10 + 15 ) = 20.
Therefore, by Heron's formula, the area is given by Area = s ( s a ) ( s b ) ( s c ) = 20 ( 5 ) ( 5 ) ( 10 ) = 50 2

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