### Video Transcript

Rectangle π΄π΅πΆπ· has π΄π΅ equals
25 and π΅πΆ equals 36. Suppose perpendiculars π΅π» and
π΄π are both of length 27. What is the area of πΆπ·ππ»?

The first thing to do in a question
like this is to sketch a diagram. Weβre told that we have a
rectangle. And coming from this will be two
perpendiculars. Therefore, weβll be working in
three dimensions. So it might be useful to draw our
rectangle like this, ready to add in the perpendiculars. Our first perpendicular starts at
point π΅ and goes up to a point π». Our second perpendicular starts at
π΄ and goes up to a point π. Weβre told that these
perpendiculars are both of length 27. So our perpendiculars should be
roughly the same length. We can then add in the dimensions
to our diagram.

Itβs worth noting that there are a
number of different diagrams we couldβve drawn. What weβre really looking for is
something to help us with our calculations. Here, weβre asked to find the area
of πΆπ·ππ». πΆπ·ππ» would look like this on
our diagram, and it would be a rectangle. In order to find out the area of
this rectangle, πΆπ·ππ», we would need to know the length and the width, which we
could multiply together to find the area. The width here would be equal to
the line π΄π΅, which means it will be 25 units long. What we do need to work out here is
the length, the line ππ·.

As we know that ππ΄ is a
perpendicular, that means that we have a right triangle in ππ΄π·. We can therefore apply the
Pythagorean theorem, which tells us that the square on the hypotenuse is equal to
the sum of the squares on the other two sides. Itβs often helpful to draw out the
triangles that weβre going to use. We know that the length ππ΄ is 27
and the length π΄π· will be the same as the length π΅πΆ, 36. We want to find the length ππ·,
which we can define as any length, but here we can call it π₯. Filling in the values into the
Pythagorean theorem, we have the hypotenuse as π₯ and the two shorter sides of 27
and 36. And it doesnβt matter which way
round we write those. We can evaluate our calculation as
π₯ squared equals 1296 plus 729. So π₯ squared equals 2025. To find π₯, we take the square root
of both sides, so π₯ equals the square root of 2025.

Usually, we keep our answer in this
square root form. But actually, 2025 is a square
number. So π₯ would be 45 units. Now, weβve calculated this length
ππ· as 45. We can go ahead and work out the
area. Multiplying our values 45 and 25,
which give us that the area of πΆπ·ππ» is equal to 1125. We werenβt given any units in the
question, but if we needed to give units here, they would of course be square units
for the area.