Video Transcript
Given that angle 𝐴 is bisected by
line segment 𝐷𝐴, that 𝐴𝐵 equals 38, 𝐴𝐶 equals 18, and 𝐵𝐶 equals 28,
determine the lengths of 𝐷𝐵 and 𝐷𝐶.
The first thing we can do is take
the information we’re given and write it down in our figure. 𝐴𝐵 equals 38. 𝐴𝐶 equals 18. We need to be careful here because
we have the distance 𝐵𝐶 and 𝐵𝐶 equals 28. At this point, it might seem like
there’s very little else we can say. But because we know that line
segment 𝐷𝐴 bisects angle 𝐴, we can use 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 adjacent sides of the triangle.
We have the opposite sides 𝐷𝐶 and
𝐷𝐵 and the adjacent sides 𝐴𝐵 and 𝐴C. Because these side lengths are in
proportion to one another, it means the ratio of opposite to adjacent will be equal
for both sets. 𝐴𝐶 over 𝐷𝐶 must be equal to
𝐴𝐵 over 𝐷𝐵. We know that 𝐴𝐶 equals 18 and
𝐴𝐵 equals 38. But this equation on its own is not
enough to solve for 𝐷𝐵 and 𝐷𝐶. So, we’ll need to write a second
equation. We know that line 𝐵𝐶 equals 28
and line 𝐵𝐶 is made up of two segments: 𝐷𝐵 and 𝐷𝐶. So, we can say that 𝐷𝐵 plus 𝐷𝐶
equals 28. And now, we have two equations that
we can use to solve for our missing side lengths.
We can look at our second equation
and try and solve for 𝐷𝐵. That means get 𝐷𝐵 by itself. To do that, we would subtract 𝐷𝐶
from both sides of the equation. And then, we could say that 𝐷𝐵
equals 28 minus 𝐷𝐶. Now that we know what 𝐷𝐵 is equal
to, we can take this information and plug it back in to our first equation. If 18 over 𝐷𝐶 equals 38 over 𝐷𝐵
and 𝐷𝐵 equals 28 minus 𝐷𝐶, then we plug in 28 minus 𝐷𝐶 for the denominator
here. But we now have two 𝐷𝐶 variables
in the denominator. And we’d like to move these to the
numerator. To do that, we cross multiply the
numerators and the denominators. 18 times 28 minus 𝐷𝐶 is equal to
38 times 𝐷𝐶. On the left, we’ll need to
distribute this times 18. 18 times 28 equals 504 and 18 times
negative 𝐷𝐶 equals negative 18 𝐷𝐶.
Now, we have 504 minus 18 𝐷𝐶
equals 38 𝐷𝐶. We want to get 𝐷𝐶 on the same
side of the equation. And so, we add 18 𝐷𝐶 to both
sides. 38 plus 18 equals 56. So we have 504 equals 56 𝐷𝐶. And then, we divide both sides of
the equation by 56 504 divided by 56 equals nine and 56 𝐷𝐶 divided by 56 equals
𝐷𝐶. And that means 𝐷𝐶 equals
nine. We can go ahead and add this to our
diagram. 𝐷𝐶 equals nine. That’s the same thing as 𝐶𝐷
equals nine. It doesn’t matter which order you
say the endpoints. Because we know that 𝐷𝐶 equals
nine, we can say that 𝐷𝐵 equals 28 minus nine. 28 minus nine is 19. So, we can say that 𝐷𝐵 equals
19.
Let’s perform a few checks
here. First, does nine plus 19 equal
28? It does. And secondly, let’s check these
proportional relationships. Is 18 over nine equal to 38 over
19? If we divide 18 by nine, we get
two. And if we divide 38 by 19, we also
get two. Two is equal to two. And so, these relationships are in
proportion with each other. And we can say that 𝐷𝐵 equals 19
and 𝐷𝐶 equals nine.
