A triangle has corners at points A, B, and C. Side AB has a length of #15 #. The distance between the intersection of point A's angle bisector with side BC and point B is #9 #. If side AC has a length of #24 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #15  #.  The distance between the intersection of point A's angle bisector with side BC and point B is #9  #.  If side AC has a length of #24  #, what is the length of side BC?

Sadie Adams · Accepted Answer

Length of side BC = 23.4

Let the point where the angle bisector intersects with side BC be D #"using the "color(blue)"angle bisector theorem"# #(AB)/(AC)=(BD)/(DC)# #15 / 24 = 9 / (DC)# #DC = (9*24) / 15 = 14.4# #BC = BD+DC= 9+14.4 =23.4#

Sophia Alfred · Answer

undefined To find the length of side BC, we can use the angle bisector theorem. According to this theorem, in a triangle, the ratio of the lengths of the two segments of a side divided by the length of the entire side is equal to the ratio of the lengths of the other two sides of the triangle divided by their lengths.

Let's denote the length of BC as $x$. According to the angle bisector theorem, we have:

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

where BD is the length of the segment of side BC adjacent to point B, and DC is the length of the segment of side BC adjacent to point C.

Given that AB = 15, AC = 24, and BD = 9, we can solve for DC:

$$ \frac{9}{x - 9} = \frac{15}{24} $$

Cross multiply and solve for $x$:

$$ 15(x - 9) = 9 \times 24 $$

$$ 15x - 135 = 216 $$

$$ 15x = 216 + 135 $$

$$ 15x = 351 $$

$$ x = \frac{351}{15} $$

$$ x = 23.4 $$

So, the length of side BC is approximately 23.4 units.