Welcome to a lesson on solving for unknown values
using the triangle proportionality theorem.
Which we proved in the previous video. Let's go ahead and review the theorem.
The parallel line to one side of a triangle
intersects the two other sides.
This would be the parallel line that intersects the two sides.
Then it divides those sides proportionally.
As you can see I listed six different proportions that are true.
Which is a result of either this theorem
or that the small triangle on the large triangle are similar.
So you may want to pause the video to review these
were gonna go and take a look at our examples.
So in this problem notice that segment
BE is parallel to segments CD.
And we want to determined the length of segment BC so let's go and label this
with an x here.
So this example is pretty straightforward.
This parallel lines cuts these two sides of the triangle.
So that the segments are proportional.
Which means that x to 12.
Would be equal to ten to 15.
So now to solve for x we can cross multiply.
15 times x would be 15x.
Must equals 12 times 10 that's
120.
Now we divide both sides by 15.
So we have x equals 120 divided by 15
equals eight.
So the length of BC
is equal to eight units.
Now let's go and take a look at a second example that is a little bit more involved.
Again here we see that segment AB is parallel to segment ED.
And were ask to find the length of AC
the length of BD
the length of DC
and the length of DE.
And the length of AC would be the length of this long segment here.
So the length of AC would just be
18 units
plus
27 units
which is equal to 45 units.
Now these next three would take a little bit more work.
In order to find the length of BD.
I'm gonna go ahead and label it x.
We can set up a proportion comparing
BD to
BC
and AE to AC.
So we'll have x to 36.
Must be equal to 18
to member AC was 45.
And now again we'll cross multiply.
So we have x times 45 that's 45x.
Must equal
36 times 18.
Which is 648.
Now we divide both sides by 45.
And I went ahead and used the calculator that's gonna be
fourteen point four units.
So the length of BD is fourteen point four.
Now determine the length of DC we have a couple options here.
Let's go ahead and use
the length of BD which we just found is fourteen point four.
So fourteen point four to y.
Must equal
18 to 27.
Again we'll cross multiply
18 times y would be 18 y.
Must equal
fourteen point four times 27.
That's gonna give us three hundred eighty-eight point eight.
And now we'll divide both sides by 18.
And y is equal to twenty-one point six.
Therefore the length of DC is equal to
twenty-on points six units.
Now I do want to mention this is not the only proportion that we could use to determine the length of DC.
Which we label y.
We could have used a proportion
y to 36
must equal 27.
to AC which is 45.
And the result of course would be the same.
The last likely to determine is a length of DE.
Which is the length of this
parallel segment.
Let's go ahead and call it z.
And this small triangle and this large triangle
are actually similar triangles.
Which means their sides are proportional.
So we could use the proportion
z to 28.
Must equal
27.
to AC which was 45 units.
And again now we cross multiply 45z.
Must equal
28 times 27.
Which is equal to 756.
And we divide both sides by 45.
756 divided by 45.
is sixteen point eight.
So the length of DE is sixteen point eight units.
And that's gonna do it for this video.
I hope you found these examples helpful.