# Video: AQA GCSE Mathematics Higher Tier Pack 2 β’ Paper 1 β’ Question 10

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 (1/2) (π + π) β. b) Given that π : π = 1 : 3 and π : π = 1 : 2, calculate the perimeter of the parallelogram in terms of π.

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### 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.

Weβre 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. Itβs 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 itβs 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. Itβs π plus π.

We said the area was a base multiplied by height, so thatβs π plus π multiplied by β. And since we try to avoid using multiplication symbols in algebra, we can write this as π plus πβ. 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. Thatβs 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.

Weβre looking now to form an expression in terms of π. So weβre going to need to do two things. Weβre going to need to form an equation for π in terms of π and an equation for π in terms of π.

Letβs 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. Weβre going to divide one by the other. Weβre going to divide π by π. Thatβs 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.

Letβs repeat this with π. This time, π to π is the same as one to two. So if we divide π by π, thatβs 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.