A triangle has corners at points A, B, and C. Side AB has a length of #32 #. The distance between the intersection of point A's angle bisector with side BC and point B is #24 #. If side AC has a length of #27 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #32  #.  The distance between the intersection of point A's angle bisector with side BC and point B is #24  #.  If side AC has a length of #27 #, what is the length of side BC?

Claire Akin · Accepted Answer

Length of side BC = 44.25

Let the point where the angle bisector intersects with side BC be D #"using the "color(blue)"angle bisector theorem"# #(AB)/(AC)=(BD)/(DC)# #32 / 27 = 24 / (DC)# #DC = (24*27) / 32 = 20.25# #BC = BD+DC= 24+20.25 =44.25#

James Atkins · Answer

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

Let's denote the length of side BC as x. According to the Angle Bisector Theorem:

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

Given that AB = 32, AC = 27, and BD = 24, we can set up the equation:

$$ \frac{32}{27} = \frac{24}{x} $$

Cross-multiplying, we get:

$$ 32x = 27 \times 24 $$
$$ 32x = 648 $$

Dividing both sides by 32, we find:

$$ x = \frac{648}{32} $$
$$ x = 20.25 $$

Therefore, the length of side BC is 20.25 units.