A triangle has corners at points A, B, and C. Side AB has a length of #36 #. 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 #42 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #36  #.  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 #42 #, what is the length of side BC?

Savannah Aguilar · Accepted Answer

#17.3333#

Let the intersection point be D (see diagram)

Using Laws of sine :

a) triangle ABD

#36/siny=8/sinx => sinx/siny=8/36 = 2/9# ----- (1)

b) triangle ADC

Let E be the distance between D and C

#42/sin(180-y) = E/sinx#

Recall that #sin(180-y)=siny#

#=> E/42 = sinx/siny# --- (2)

Eq (1) = Eq (2)

#=> E/42 =2/9#

#=> E = (42xx2)/9 =9.3333#

Length of BC #= 8+9.3333=17.3333#

Wyatt Anderson · Answer

undefined To find the length of side BC, we can use the angle bisector theorem.

Given:
AB = 36
AC = 42
Distance from the intersection of point A's angle bisector with side BC to point B = 8

Let's denote the length of side BC as x.

According to the angle bisector theorem, the ratio of the lengths of the two segments of a side bisected by an angle bisector is equal to the ratio of the lengths of the other two sides of the triangle. Mathematically, this can be represented as:

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

Where BD is the length from point B to the intersection of the angle bisector with side BC, and DC is the length from that intersection to point C.

Given:
AB = 36
AC = 42
BD = 8
DC = x - 8 (because the whole length BC is x)

Plugging in the given values:

$$ \frac{36}{42} = \frac{8}{x - 8} $$

Simplifying:

$$ \frac{6}{7} = \frac{8}{x - 8} $$

Cross-multiplying:

$$ 6(x - 8) = 7(8) $$

Expanding:

$$ 6x - 48 = 56 $$

Adding 48 to both sides:

$$ 6x = 104 $$

Dividing both sides by 6:

$$ x = \frac{104}{6} $$

$$ x = 17.33 $$

So, the length of side BC is approximately 17.33 units.