### Video Transcript

Trapezoid π΄π΅πΆπ· has vertices π΄:
four, 14; π΅: four, negative four; πΆ: negative 12, negative four; and π·: negative
12, nine. Given that vector ππ is parallel
to vector ππ and vector ππ is perpendicular to vector ππ, find the area of
that trapezoid.

Letβs consider what we are told
about the trapezoid. We know that vector ππ is
parallel to vector ππ. We also know that vector ππ is
perpendicular to vector ππ. This means that they meet at right
angles. It therefore follows that vector
ππ also meets vector ππ at right angles. We know that the area of a
trapezoid can be found using the formula π plus π over two multiplied by β, where
π and π are the parallel sides and β is the perpendicular height. We need to find the length of the
sides π΄π΅ and π·πΆ and the perpendicular height πΆπ΅.

In order to calculate the length of
the sides, we need to work out the magnitude of the vectors. We will begin by calculating the
magnitude of ππ. This is equal to the square root of
four minus four squared plus negative four minus 14 squared. Four minus four is equal to
zero. And negative four minus 14 is
negative 18. Squaring this gives us 324, and
then square rooting the answer gives us 18. The magnitude of vector ππ is
18. We can repeat this process to
calculate the magnitude of vector ππ. This is equal to the square root of
negative 12 minus negative 12 squared plus negative four minus nine squared. This gives us an answer of 13.

As the magnitude of ππ is greater
than the magnitude of ππ, we can see that our sketch has not been drawn to
scale. It would therefore make more sense
to relabel it as shown. Vector ππ is still parallel to
vector ππ, and vector ππ is perpendicular to vector ππ. We can now add the lengths onto our
diagram. We now need to calculate the
magnitude of vector ππ. Using the same method, we see that
this is equal to 16. We now have the lengths of the
parallel sides as well as the length of the perpendicular height of the
trapezoid. The area is therefore equal to 13
plus 18 divided by two multiplied by 16. 13 plus 18 is equal to 31. Multiplying 31 over two by 16 gives
us 248. The area of the trapezoid is
therefore equal to 248 square units.