A triangle has corners at points A, B, and C. Side AB has a length of #16 #. The distance between the intersection of point A's angle bisector with side BC and point B is #8 #. 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 #16  #.  The distance between the intersection of point A's angle bisector with side BC and point B is #8  #.  If side AC has a length of #27 #, what is the length of side BC?

Allison Allen · Accepted Answer

Length of side BC = 21.5

Let the point where the angle bisector intersects with side BC be D #"using the "color(blue)"angle bisector theorem"# #(AB)/(AC)=(BD)/(DC)# #16 / 27 = 8 / (DC)# #DC = (8*27) / 16 = 13.5# #BC = BD+DC= 8+13.5 =21.5#

Wyatt Anderson · Answer

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

Let's denote the intersection point of the angle bisector from point A with side BC as D. According to the problem, BD = 8 units.

Given:
- AB = 16 units
- AD is the angle bisector of angle A.
- AC = 27 units
- BD = 8 units

By the Angle Bisector Theorem, the ratio of the length of the two segments into which side BC is divided by the angle bisector (i.e., BD to DC) is equal to the ratio of the lengths of the other two sides of the triangle (i.e., AB to AC).

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

Substituting the given values, we get:

$\frac{8}{DC} = \frac{16}{27}$

Solving for $DC$ gives:

$DC = \frac{8 \times 27}{16} = \frac{216}{16} = 13.5$ units

Therefore, the length of side BC is the sum of BD and DC, which is $8 + 13.5 = 21.5$ units.