A triangle has corners at points A, B, and C. Side AB has a length of #18 #. The distance between the intersection of point A's angle bisector with side BC and point B is #3 #. If side AC has a length of #24 #, what is the length of side BC?

Question

A triangle has corners at points A, B, and C.  Side AB has a length of #18  #.  The distance between the intersection of point A's angle bisector with side BC and point B is #3  #.  If side AC has a length of #24 #, what is the length of side BC?

Theodore Abad · Accepted Answer

#BC=7# #"let D be the point where the angle bisector meets BC"# #"using the "color(blue)"angle bisector theorem"# #color(red)(bar(ul(|color(white)(2/2)color(black)((AB)/(AC)=(BD)/(DC))color(white)(2/2)|)))# Require to find DC. #rArr18/24=3/(DC)# #color(blue)"cross-multiplying"# gives. #18DC=3xx24# #rArrDC=(3xx24)/18=4# #BC=BD+DC=3+4=7#

Sadie Allen · Answer

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

Let's denote the length of side BC as x. Since the angle bisector from point A divides side BC into two segments, let's call the length of the segment from point A to the intersection as y, and the length of the segment from the intersection to point C as z.

According to the Angle Bisector Theorem:

y/z = AB/AC

Substituting the given values:

y/z = 18/24 = 3/4

Now, we know that the length of the segment from the intersection to point B is 3. So, we can set up another equation:

y + 3 = x

Now, we have a system of two equations:

1. y/z = 3/4
2. y + 3 = x

From equation 1, we can express y in terms of z:

y = (3/4)z

Substituting this expression for y into equation 2:

(3/4)z + 3 = x

Now, we can solve for z:

(3/4)z + 3 = x
(3/4)z = x - 3
z = (4/3)(x - 3)

We also know that y + z = x:

(3/4)z + z = x
(3/4)z + (4/3)(x - 3) = x

Now, we can solve this equation for x:

(3/4)(4/3)(x - 3) + (4/3)(x - 3) = x

After solving this equation, we'll find the value of x, which represents the length of side BC.

James Atkins · Answer

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

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

According to the Angle Bisector Theorem, we have:

$$
\frac{{AC}}{{AB}} = \frac{{BC}}{{AB}} = \frac{{AC + AB}}{{BC + AB}}
$$

Given that $ AB = 18 $, $ AC = 24 $, and the distance from the intersection of the angle bisector at point A to point B is 3, we can substitute these values into the equation:

$$
\frac{{24}}{{18}} = \frac{{18 + 24}}{{x + 18}}
$$

Now, let's solve for $ x $:

$$
\frac{{24}}{{18}} = \frac{{18 + 24}}{{x + 18}}
$$

$$
\frac{4}{3} = \frac{42}{x + 18}
$$

$$
4(x + 18) = 3 \times 42
$$

$$
4x + 72 = 126
$$

$$
4x = 126 - 72
$$

$$
4x = 54
$$

$$
x = \frac{54}{4}
$$

$$
x = 13.5
$$

Therefore, the length of side BC is $ \boxed{13.5} $.