A triangle has corners at points A, B, and C. Side AB has a length of #21 #. The distance between the intersection of point A's angle bisector with side BC and point B is #7 #. If side AC has a length of #28 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #21  #.  The distance between the intersection of point A's angle bisector with side BC and point B is #7  #.  If side AC has a length of #28 #, what is the length of side BC?

James Atkins · Accepted Answer

Length of side BC = 16.3333

Let the point where the angle bisector intersects with side BC be D #"using the "color(blue)"angle bisector theorem"# #(AB)/(AC)=(BD)/(DC)# #21 / 28 = 7 / (DC)# #DC = (7*28) / 21 = 9.3333# #BC = BD+DC= 7+9.3333 =16.3333 #

Savannah Aguilar · Answer

undefined To find the length of side BC, we can use the Angle Bisector Theorem, which states that in a triangle, the length of the side opposite an angle bisector is proportional to the lengths of the other two sides.

Let $ D $ be the point where the angle bisector of angle $ A $ intersects side $ BC $.

According to the Angle Bisector Theorem:

$$ \frac{BD}{DC} = \frac{AB}{AC} $$

Given that $ AB = 21 $, $ AC = 28 $, and $ BD = 7 $, we can plug these values into the theorem and solve for $ DC $:

$$ \frac{7}{DC} = \frac{21}{28} $$

$$ 7DC = 21 \times 28 $$

$$ 7DC = 588 $$

$$ DC = \frac{588}{7} $$

$$ DC = 84 $$

Therefore, the length of side BC is $ BD + DC = 7 + 84 = 91 $. So, the length of side BC is 91.