Video Transcript
In triangle 𝐴𝐵𝐶, let 𝐷 be a
point on the line segment 𝐴𝐵 and 𝐸 a point on the line segment 𝐴𝐶. Suppose that 𝐴𝐷 over 𝐷𝐵 equals
three over two and that the line segment 𝐷𝐸 is parallel to the line segment
𝐵𝐶. If the area of the triangle 𝐴𝐵𝐶
is 237, what is the area of the trapezoid 𝐷𝐵𝐶𝐸?
Well, what we’re dealing with is in
fact two similar triangles. First of all, we had the triangle
𝐴𝐷𝐸, and then we have the larger triangle 𝐴𝐵𝐶. And we know that triangle 𝐴𝐵𝐶 is
similar to triangle 𝐴𝐷𝐸, as angle 𝐴 is a shared angle — that’s the angle at 𝐴 —
and the angle 𝐴𝐷𝐸 is equal to angle 𝐴𝐵𝐶 as they are corresponding angles. And that’s because we have parallel
lines 𝐷𝐸 and 𝐵𝐶. And this is using the proof which
is angle-angle. And in fact, we know that there’ll
be a third angle the same, again because of corresponding angles, and those will be
angles 𝐴𝐸𝐷 and angle 𝐴𝐶𝐵.
Okay, great. Now that we know that they are
similar, we can use their properties to help solve this problem. And that’s because similar
triangles have corresponding lengths in proportion. And we also know with similar
triangles that one is going to be an enlargement of the other. And the method that we’re going to
use to solve this problem is the scale factor method.
Well, first of all, what we can do
is use a bit of information we’ve been given, and that is that 𝐴𝐷 over 𝐷𝐵 is
equal to three over two. Well, what we want to have, in
fact, is 𝐴𝐷 over 𝐴𝐵. And that’s because that’s looking
at the two corresponding sides of our two triangles. Well, this is going to be equal to
three over five. And that’s because we’ve still got
𝐴𝐷, so that’s three parts, and 𝐴𝐵 is gonna be three plus two parts, which is
five parts. So, we get three over five.
It is worth mentioning at this
point, though, these are not in fact necessarily the lengths of 𝐴𝐷 and 𝐷𝐵; these
are just the parts in our ratio. Now, if we multiply both sides by
𝐴𝐵, what we get is that 𝐴𝐷 is equal to three-fifths multiplied by 𝐴𝐵. So therefore, we can say that the
linear scale factor — we call it the linear scale factor because that’s looking at
our sides, so our dimensions here — is going to be equal to three-fifths.
And this is when we go from the
larger triangle, triangle 𝐴𝐵𝐶, to the smaller triangle, triangle 𝐴𝐷𝐸. And we’re looking for the scale
factor in this direction because the bit of information that we’re given is the area
of triangle 𝐴𝐵𝐶. And what we’re going to want to do
is find the area of triangle 𝐴𝐷𝐸.
Well, as we’re looking at areas,
then what we want to do now is think about something that we know, and that’s a
relationship between the linear scale factor and the area scale factor. And that is, that the linear scale
factor is equal to the area scale factor squared. So, therefore, our area scale
factor from triangle 𝐴𝐵𝐶 to triangle 𝐴𝐷𝐸 is going to be equal to three-fifths
squared, which is gonna be equal to nine over 25.
So now what we can do is calculate
the area of triangle 𝐴𝐷𝐸 using this area scale factor. But why do we want to do this? Well, we’re looking to find the
area of the trapezoid 𝐷𝐵𝐶𝐸. And this can be found by
subtracting the area of triangle 𝐴𝐷𝐸 from the area of triangle 𝐴𝐵𝐶. So, the area of triangle 𝐴𝐷𝐸 is
going to be equal to 237 multiplied by nine over 25. And this is gonna be equal to
85.32.
So, now, we can move on to find the
area of the trapezoid 𝐷𝐵𝐶𝐸, which I’ve shaded here in blue. And as we already stated, this can
be found by subtracting the area of the triangle 𝐴𝐷𝐸 from the area of the
triangle 𝐴𝐵𝐶. So, it’s 237 minus 85.32, which is
gonna give us an area of the trapezoid 𝐷𝐵𝐶𝐸 of 151.68.
