Video Transcript
Use determinants to work out the
area of the triangle with vertices two, negative two; four, negative two; and zero,
two by viewing the triangle as half a parallelogram.
We want to find out the area of a
triangle by using determinants. And we’re given the coordinates of
its vertices. So we could just do this directly
from our formula. However, the question doesn’t want
us to do this. It wants us to do this by viewing
the triangle as half of a parallelogram. So to do this, we’ll start by
sketching our three points to get an idea of what our parallelogram could look
like. If we plot the three points and
connect them, we get a triangle which looks like this. The question then becomes, how are
we going to turn this into a parallelogram?
And although it may not seem like
it, there’s actually three different ways we could do this. There’s a few different ways of
seeing this. One way to do this is
geometrically. Let’s draw an exact copy of our
triangle. And we can consider the shape if we
were to glue two of the sides together. If we were to do this, we would get
a shape which looks like this. And we could find the coordinate of
this vertex by using what we know about vectors. This is a parallelogram with twice
the area of our original triangle because it’s made up of two triangles of equal
area. But this wasn’t the only choice for
the side. For example, we could’ve chosen
this side. If we were to glue these two
triangles together along this edge, we would get something which looks like the
following diagram. And once again, we could find the
coordinates of this vertex by using what we know about vectors.
Once again, the area of this
triangle plus the green triangle is still twice the area of our original
triangle. So it’s a parallelogram with twice
the area. Finally, we could do exactly the
same thing by combining the last two edges. And if we did this, we would get a
similar story. We get a third parallelogram. We can find the coordinates of its
vertex. And this parallelogram is twice the
area of our original triangle. It doesn’t matter which of these we
choose. For simplicity, we’ll choose the
following example. And it’s worth pointing out we
didn’t need to do this geometrically. You can also do this by choosing
two of the sides of the triangle as a vector. This would give us the same
result.
The question now wants us to find
the area of this parallelogram. And it wants us to do this by using
determinants. Recall, we know how to find the
area of a parallelogram by using determinants. If we define our parallelogram by
the vectors of its sides, 𝐕 one and 𝐕 two, then its area 𝐴 is equal to the
absolute value of the determinant of the two-by-two matrix 𝑎, 𝑏, 𝑐, 𝑑, where our
row vectors are the rows of our matrix. And we could’ve also used column
vectors and used the columns of our matrix instead. It wouldn’t change our answer.
So to answer this question, we need
to find the vectors 𝐕 one and 𝐕 two. There’s a few different ways of
doing this. Let’s start by finding 𝐕 one. In 𝐕 one, our 𝑥-component starts
at two and ends at zero. So the horizontal component or
change in 𝑥 of 𝐕 one is zero minus two, which is of course just negative two. We can do the same for the vertical
component. We end at a 𝑦-coordinate of two
and begin at negative two. So we get two minus negative two,
which is four. So 𝐕 one is the vector negative
two, four. We can then do exactly the same to
find vector 𝐕 two. 𝐕 two is two, zero.
We’re now ready to use our
formula. The area of our parallelogram is
the absolute value of the determinant of the two-by-two matrix negative two, four,
two, zero, where negative two, four is 𝐕 one and two, zero is 𝐕 two. And we can evaluate this
expression. We get the absolute value of
negative two times zero minus four times two, which is just equal to eight. But remember, this is the area of
our parallelogram. This is twice the area of our
triangle. So we need to divide this value by
two. So we divide through by two. And we get the area of our triangle
is equal to four.
And one thing worth pointing out
here is we never needed to calculate the fourth coordinate of our parallelogram. This is because the area of a
parallelogram is entirely defined by the vectors which make up the
parallelogram.
