Triangle angle bisector problem: Finding area

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

Question

Triangle angle bisector problem: Finding areaArea   △  A  B  C=?How to apply this theorem?

Douglas Woodard · Accepted Answer

Step 1Area   △  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 2These equation have only one valid solution, namely,   x  =  9  ,  y  =  5  ,  z  =  6.Hence the sides of the triangle, are 15,10,15the semi-perimeter is   s  =      1    2    (  15  +  10  +  15  )  =  20.Therefore, by Heron&#039;s formula, the area is given by   Area  =      s    (    s    −    a    )    (    s    −    b    )    (    s    −    c    )    =      20    (    5    )    (    5    )    (    10    )    =  50      2

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

Open question

Answer & Explanation

New Questions in High school geometry