### Video Transcript

Given that ๐ด๐ท over ๐ท๐ถ equals
three over two and the area of triangle ๐ด๐ต๐ถ equals 695 centimeters squared, find
the area of trapezoid ๐ท๐ถ๐ต๐ธ.

So in this problem, what weโre
looking at are two similar triangles. Weโve got the triangle ๐ด๐ท๐ธ and a
triangle ๐ด๐ต๐ถ. And when we are dealing with
similar triangles, then what this means is that we have one triangle which is an
enlargement or dilation of the other. So they are in fact in proportion
and have all the corresponding angles equal.

And we can prove that in this
problem because first of all we have one shared angle at ๐ด. And then what we also know is
because we have two parallel lines, which are denoted here by these arrowheads, and
thatโs the two parallel lines ๐ท๐ธ and ๐ถ๐ต, then in fact the angle ๐ด๐ธ๐ท is going
to be equal to the angle ๐ด๐ต๐ถ because these are corresponding angles.

So therefore, we can say that
triangle ๐ด๐ธ๐ท is similar to triangle ๐ด๐ต๐ถ. And weโve done that using the
angle-angle proof. And thatโs because if we have two
angles the same, then the third angle must be the same. And thatโs because all the angles
in a triangle add up to 180. And in fact in our problem, we know
that the two angles are going to be the same โ thatโs angle ๐ด๐ท๐ธ and angle ๐ด๐ถ๐ต
โ because once again theyโre corresponding angles.

Okay, great. So we now know the properties
between our triangles, and that is that they are similar. Well, as the two triangles are
similar to each other, we know that one is going to be an enlargement of the
other. So the method weโre going to use to
solve this problem is scale factor. So weโre gonna find the scale
factor of enlargement.

So before we do that, letโs take a
look at the information we have. So we know that ๐ด๐ท over ๐ท๐ถ is
equal to three over two. So therefore, what we can say is
that ๐ด๐ท over ๐ด๐ถ is going to be equal to three over five. And thatโs because if we think
about it, ๐ด๐ท is represented by our three and our ๐ท๐ถ is represented by our
two. And what we must remember that this
is not necessarily the length of ๐ด๐ท and ๐ท๐ถ. Itโs just what itโs represented by
in our fraction or ratio that we have. Well, therefore, if three parts is
๐ด๐ท and two parts is ๐ท๐ถ, then ๐ด๐ถ must be three plus two, which is five
parts. So therefore, we can say that ๐ด๐ท
over ๐ด๐ถ is gonna be equal to three over five.

So then what we can do is multiply
through by ๐ด๐ถ. When we do that, we get ๐ด๐ท is
equal to three-fifths multiplied by ๐ด๐ถ. And then if we divide by
three-fifths, what we would get is that five-thirds multiplied by ๐ด๐ท โ and thatโs
because if you divide by three over five, itโs the same as multiplying by five over
three โ is going to be equal to ๐ด๐ถ. So what we can say is that to get
๐ด๐ถ from ๐ด๐ท, and thatโs a length on the small triangle to a length on the big
triangle, we multiply by five over three, or five-thirds. So therefore, we can say that the
scale factor of enlargement or dilation is going to be five over three, or
five-thirds.

Now, in this problem, weโre not
actually looking to find a length of one of the sides or a length of our
trapezoid. What weโre looking to deal with is
area. So therefore, what we need to do is
actually find the area scale factor. And what we know about the area
scale factor is that the area scale factor is equal to the linear scale factor
squared. So therefore, our area scale factor
is gonna be equal to five-thirds squared, which is equal to 25 over nine.

So therefore, what we can say is
that the area of our triangle ๐ด๐ท๐ธ โ so this is the area of the smaller triangle,
so weโre going from the bigger triangle to the smaller triangle โ is going to be
equal to the area of the bigger triangle ๐ด๐ต๐ถ, which is 695, divided by our scale
factor, our area scale factor in fact. And that is gonna be 25 over nine,
which is gonna give us 250.2, and that would be centimeters squared.

Okay, have we solved the
problem? Well no, this is not in fact what
weโre looking to find. In fact, what weโre looking to find
is the area of trapezoid ๐ท๐ถ๐ต๐ธ, which Iโve shaded here in blue. And to find this, what weโre going
to do is subtract the area of the smaller triangle ๐ด๐ท๐ธ away from the area of the
larger triangle ๐ด๐ต๐ถ. So weโre going to have 695 minus
250.2, which is going to be equal to 444.8. So therefore, we can say that the
area of trapezoid ๐ท๐ถ๐ต๐ธ is going to be 444.8 centimeters squared.