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‘Ž, 𝑏, and β„Ž, show that the formula for the area of the trapezium is (1/2) (π‘Ž + 𝑏) β„Ž. b) Given that π‘Ž : 𝑏 = 1 : 3 and π‘Ž : 𝑐 = 1 : 2, calculate the perimeter of the parallelogram in terms of π‘Ž.

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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‘Ž, 𝑏, and β„Ž, show that the formula for the area of the trapezium is one-half multiplied by π‘Ž plus 𝑏 multiplied by β„Ž.

We’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. It’s simply β„Ž. 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 it’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‘Ž, and the length of the longer parallel side is 𝑏. This means we can form an expression for the base of the parallelogram. It’s π‘Ž plus 𝑏.

We said the area was a base multiplied by height, so that’s π‘Ž plus 𝑏 multiplied by β„Ž. And since we try to avoid using multiplication symbols in algebra, we can write this as π‘Ž plus π‘β„Ž. 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. That’s a half multiplied by π‘Ž plus 𝑏 multiplied by β„Ž.

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

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‘Ž, 𝑏, and 𝑐. And we can form an expression for the perimeter as two π‘Ž plus two 𝑏 plus two 𝑐.

We’re looking now to form an expression in terms of π‘Ž. So we’re going to need to do two things. We’re going to need to form an equation for 𝑏 in terms of π‘Ž and an equation for 𝑐 in terms of π‘Ž.

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

Let’s repeat this with 𝑐. This time, π‘Ž to 𝑐 is the same as one to two. So if we divide 𝑐 by π‘Ž, that’s the same as dividing two by one, which is simply two. We can form an equation for 𝑐 in terms of π‘Ž by multiplying both sides by π‘Ž again. And we get 𝑐 is equal to two π‘Ž.

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

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