Video Transcript
In the figure, line 𝐴𝐷 bisects
angle 𝐵𝐴𝐶, 𝐵𝐷 equals eight, 𝐷𝐶 equals 11, and the perimeter of the triangle
𝐴𝐵𝐶 is 57. Determine the lengths of line 𝐴𝐵
and line 𝐴𝐶.
First, let’s add the information we
know to the figure. Since line 𝐴𝐷 bisects angle
𝐵𝐴𝐶, angle 𝐶𝐴𝐷 is equal to angle 𝐵𝐴𝐷. 𝐵𝐷 equals eight. 𝐷𝐶 equals 11. The perimeter of this triangle is
57. That means the distance all the way
around the triangle is 57, which would be eight plus 11 plus 𝐴𝐶 plus 𝐴𝐵.
At this point, it might seem like
we don’t have enough information to solve this problem. But we know the angle bisector
theorem, which tells us that an angle bisector of a triangle divides the opposite
side into two segments that are proportional to the other two adjacent sides. From angle 𝐶𝐴𝐷, it has an
opposite side and an adjacent side. From angle 𝐵𝐴𝐷, we have the
opposite and adjacent sides. And the side lengths — the opposite
over adjacent — should be equal to one another. The ratio of side lengths 𝐶𝐷 to
𝐴𝐶 will be equal to 𝐵𝐷 over 𝐴𝐵. We know that 𝐶𝐷 equals 11 and
that 𝐵𝐷 equals eight. But this relationship alone is not
enough to help us solve the problem. We’ll need both of these equations,
the equation for the perimeter and the equation from the relationship of the side
lengths, to solve the problem.
The first thing we wanna do is try
to get either 𝐴𝐶 or 𝐴𝐵 by itself. We can multiply 𝐴𝐵 times 11 and
𝐴𝐶 times eight. We might call this cross
multiplication. From there, we can divide both
sides of the equation by 11. On the left, it cancels out. And then, we can say that 𝐴𝐵 is
equal to eight 11ths times 𝐴𝐶. And we can take this information
and plug it in to the perimeter problem. The perimeter is 57 and eight plus
11 is 19. So we can say that 57 is equal to
19 plus 𝐴𝐶 plus 𝐴𝐵. But then, we could subtract 19 from
both sides, which tells us that 38 equals 𝐴𝐶 plus 𝐴𝐵.
Now, we’re ready to use this
information. We know what 𝐴𝐵 is equal to. And so, we plug that in. 𝐴𝐵 equals eight 11ths 𝐴𝐶. And so, we can say that 38 equals
𝐴𝐶 plus eight 11ths 𝐴𝐶. Since we have one 𝐴𝐶, we can
rewrite that as 11 over 11 𝐴𝐶. We can combine our two 𝐴𝐶 terms
to get 19 over 11 𝐴𝐶. To get 𝐴𝐶 by itself, we’ll
multiply by the reciprocal of 19 over 11, which is 11 over 19, on both sides of the
equation. 11 19ths times 38 equals 22 and 19
over 11 times 11 over 19 is one, which means we’re left with one 𝐴𝐶 or just
𝐴𝐶. And so, we can say that 𝐴𝐶 is
22. If 38 equals 𝐴𝐶 plus 𝐴𝐵 and
𝐴𝐶 equals 22, we can subtract 22 from both sides. 38 minus 22 equals 16. Therefore, 𝐴𝐵 equals 16.
Let’s go back to the ratio we set
up at the beginning. We found that 𝐴𝐶 was equal to 22
and 𝐴𝐵 was equal to 16. 11 over 22 is equal to one over
two. And eight over 16 is also equal to
one over two. This means that the angle bisector
has divided the two segments in proportion one to two for the opposite over the
adjacent. We can perform one final check by
seeing if 57 is equal to eight plus 11 plus 22 plus 16 and it is.
And so, we give our final answer:
line segment 𝐴𝐵 equals 16 and line segment 𝐴𝐶 equals 22.
