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 #12 #. If side AC has a length of #33 #, 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 #12  #.  If side AC has a length of #33 #, what is the length of side BC?

Maverick Smith · Accepted Answer

38.4

Here is a sketch (not to scale)

To calculate the length of BC we require to find the length of DC

To do this use the#color(blue)" Angle bisector theorem"#
For the triangle ABC given this is.

#color(red)(|bar(ul(color(white)(a/a)color(black)((BD)/(DC)=(AB)/(AC))color(white)(a/a)|)))#
Substitute the appropriate values into the ratio

#rArr12/(DC)=15/33#
now cross-multiply

#rArr15xxDC=33xx12rArrDC=(33xx12)/15=26.4#
Thus length of BC = BD + DC = 12 + 26.4 = 38.4

Avery Allen · Answer

undefined Using the Angle Bisector Theorem, the length of side BC can be calculated as follows:

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

According to the Angle Bisector Theorem:

BD/DC = AB/AC

Given that AB = 15, AC = 33, and BD = 12, we can find DC:

12/DC = 15/33

Solving for DC:

DC = (33 * 12) / 15
   = 396 / 15
   = 26.4

Now, BC can be found:

BC = BD + DC
   = 12 + 26.4
   = 38.4

Therefore, the length of side BC is 38.4 units.