A triangle has corners at points A, B, and C. Side AB has a length of #9 #. The distance between the intersection of point A's angle bisector with side BC and point B is #6 #. If side AC has a length of #21 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #9  #.  The distance between the intersection of point A's angle bisector with side BC and point B is #6  #.  If side AC has a length of #21 #, what is the length of side BC?

Caroline Aucoin · Accepted Answer

Length of side BC = 20

Let the point where the angle bisector intersects with side BC be D #"using the "color(blue)"angle bisector theorem"# #(AB)/(AC)=(BD)/(DC)# #9 / 21 = 6 / (DC)# #DC = (6*21) / 9 = 14# #BC = BD+DC= 6+14 =20#

Jayden Jackson · Answer

undefined Using the Angle Bisector Theorem, we can find the length of side BC. The theorem states that in a triangle, an angle bisector divides the opposite side into segments that are proportional to the lengths of the other two sides.

Given that the length of side AB is 9, and the distance between the intersection of point A's angle bisector with side BC and point B is 6, we can set up the following proportion:

$$ \frac{AC}{AB} = \frac{BC}{6} $$

Substituting the given values, we have:

$$ \frac{21}{9} = \frac{BC}{6} $$

Solving for BC:

$$ BC = \frac{21}{9} \times 6 $$
$$ BC = 14 $$

Therefore, the length of side BC is 14.