### Video Transcript

Part a) Two identical trapeziums
have been placed together to form a parallelogram. By first calculating an expression
for the area of the parallelogram in terms of π, π, and β, show that the formula
for the area of the trapezium is one-half multiplied by π plus π multiplied by
β.

Weβre told to begin this question
by finding an expression for the area of the parallelogram. Remember, the formula for area of a
parallelogram is the base multiplied by its perpendicular height. We already have an expression for
the height. Itβs simply β. We can find an expression for the
base of this parallelogram by considering the trapezium on the right-hand side of
the picture.

This trapezium is identical to the
one on the left, but itβs been placed upside down and next to it. This means it has the same
dimensions. The length of the shorter parallel
side is π, and the length of the longer parallel side is π. This means we can form an
expression for the base of the parallelogram. Itβs π plus π.

We said the area was a base
multiplied by height, so thatβs π plus π multiplied by β. And since we try to avoid using
multiplication symbols in algebra, we can write this as π plus πβ. We know that the two trapeziums are
identical in size. So we can halve the area of the
parallelogram to find the area of one of the trapeziums. Thatβs a half multiplied by π plus
π multiplied by β.

Part b) Given that π to π is
equal to one to three and π to π is equal to one to two, calculate the perimeter
of the parallelogram in terms of π.

The perimeter of a shape is the
total distance around the outside. Since the two trapeziums are
identical, we can add the following dimensions π, π, and π. And we can form an expression for
the perimeter as two π plus two π plus two π.

Weβre looking now to form an
expression in terms of π. So weβre going to need to do two
things. Weβre going to need to form an
equation for π in terms of π and an equation for π in terms of π.

Letβs begin with π. π to π is one to three. The first part of the ratio then
represents π, and the second part represents π. Weβre going to divide one by the
other. Weβre going to divide π by π. Thatβs the same as dividing three
by one. But of course, three divided by one
is simply three. So our equation is π divided by π
equals three. We multiply both sides of this
equation by π. And that tells us that π is equal
to three π. We have an equation for π in terms
of π.

Letβs repeat this with π. This time, π to π is the same as
one to two. So if we divide π by π, thatβs
the same as dividing two by one, which is simply two. We can form an equation for π in
terms of π by multiplying both sides by π again. And we get π is equal to two
π.

We now need to substitute these
equations for π and π back into the original equation for the perimeter of the
parallelogram. It becomes two π plus two lots of
π, which is two lots of three π, plus two lots of π, which is two lots of two
π. Two multiplied by three π is six
π, and two multiplied by two π is four π. So our expression becomes two π
plus six π plus four π, which is equal to 12π. And we can say that the perimeter
of the parallelogram in terms of π is 12π or 12π centimetres.