A triangle has corners at points A, B, and C. Side AB has a length of #10 #. The distance between the intersection of point A's angle bisector with side BC and point B is #2 #. If side AC has a length of #8 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #10  #.  The distance between the intersection of point A's angle bisector with side BC and point B is #2  #.  If side AC has a length of #8  #, what is the length of side BC?

James Atkins · Accepted Answer

#bc=4#

Consider the diagram

Always remember that the angle bisector of any angle in a triangle also bisects the opposite side into two equal halves

So,

We can say that

#color(purple)(bd=dc#

#rarr2=2#

So,the length of

#color(green)(bc=bd+dc=2+2=4#

Noah Atkinson · Answer

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

Given:
- Side AB has a length of 10.
- The distance between the intersection of point A's angle bisector with side BC and point B is 2.
- Side AC has a length of 8.

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

Using the angle bisector theorem, we can set up the proportion:

$$
\frac{{AC}}{{AB}} = \frac{{\text{Length of segment } AC}}{{\text{Length of segment } CB}}
$$

Plugging in the given values:

$$
\frac{{8}}{{10}} = \frac{{x + 2}}{{x}}
$$

Solving this equation for $x$:

$$
8x = 10(x + 2)
$$
$$
8x = 10x + 20
$$
$$
2x = 20
$$
$$
x = 10
$$

So, the length of side BC is $10$.