A triangle has corners at points A, B, and C. Side AB has a length of #9 #. 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 #15 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #9  #.  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 #15  #, what is the length of side BC?

Maverick Smith · Accepted Answer

#16#

Angle bisector theorem : #(AB)/(BD)=(AC)/(DC)#

Given that #AB=9, BD=6, AC=15#
#=> DC=(15xx6)/9=10#

#BC =BD+DC#
#=> BC=6+10=16#

Evelyn Arboleda · Answer

undefined Using the Angle Bisector Theorem, we can find the length of side BC.

The theorem states that in a triangle, if a line bisects an angle, it divides the opposite side into segments that are proportional to the lengths of the other two sides.

Let D be the point where the angle bisector of angle A intersects side BC. Then, according to the Angle Bisector Theorem:

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

Given that AB = 9 and AC = 15, we can substitute these values into the equation:

$\frac{BD}{CD} = \frac{9}{15} = \frac{3}{5}$

Let x be the length of BC. Then, BD = x - 6 and CD = x.

Substituting these expressions into the ratio, we get:

$\frac{x-6}{x} = \frac{3}{5}$

Cross-multiplying:

$5(x-6) = 3x$

Solve for x:

$5x - 30 = 3x$

$2x = 30$

$x = 15$

So, the length of side BC is 15 units.