A triangle has corners at points A, B, and C. Side AB has a length of #24 #. 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 #16 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #24  #.  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 #16 #, what is the length of side BC?

Wyatt Anderson · Accepted Answer

Length of side BC = 10

Let the point where the angle bisector intersects with side BC be D #"using the "color(blue)"angle bisector theorem"# #(AB)/(AC)=(BD)/(DC)# #24 / 16 = 6 / (DC)# #DC = (6*16) / 24 = 4# #BC = BD+DC= 6+4 =10#

Carson Arnold · Answer

undefined To find the length of side BC in the triangle ABC, we can use the angle bisector theorem.

Let D be the point where the angle bisector from point A intersects side BC. According to the angle bisector theorem, the ratio of the lengths of the segments formed by the angle bisector in a triangle is equal to the ratio of the lengths of the sides opposite those segments.

Given:
- Length of side AB (opposite angle at C) = 24
- Length of side AC (opposite angle at B) = 16
- Length from D to B = 6

Using the angle bisector theorem:
$$ \frac{BD}{DC} = \frac{AB}{AC} $$

Substitute the given values:
$$ \frac{6}{DC} = \frac{24}{16} $$

Solve for DC:
$$ \frac{6}{DC} = \frac{3}{2} $$
$$ DC = \frac{2}{3} \times 6 $$
$$ DC = 4 $$

Finally, to find the length of side BC:
$$ BC = BD + DC = 6 + 4 = 10 $$

Therefore, the length of side BC is 10 units.