Question 39 of 42advanced🔍 AnalyzeLong Answer5 marks

Show how we can use two identical copies of a trapezium to make a parallelogram. How will this give us the formula for the area of a trapezium?

(i)
Answer

To make a parallelogram from two identical copies of a trapezium, we take two congruent trapeziums. We rotate one trapezium by 180 degrees and fit it together with the other trapezium along a non-parallel side.

This arrangement aligns the parallel sides aa and bb to form a straight line, creating a larger parallelogram. The area of this constructed parallelogram is exactly double the area of one trapezium.

(ii)
Answer

The base of this new parallelogram is the sum of the parallel sides of the trapezium, which is (a+b)(a + b). The height of the parallelogram remains the same as the height of the trapezium, hh.

We know that the area of a parallelogram is given by base × height. Therefore, the area of our constructed parallelogram is (a+b)×h(a + b) \times h.

Since this parallelogram is made of two identical trapeziums, the area of a single trapezium is half the area of the parallelogram. Hence,

Area of Trapezium=12×(a+b)×h\text{Area of Trapezium} = \frac{1}{2} \times (a + b) \times h

Explanation

The textbook explains that two congruent copies of a shape can be fitted together to make a parallelogram (as demonstrated with triangles). By rotating one identical trapezium and joining it to another, the parallel sides aa and bb combine to form the base of a new parallelogram. Using the known formula for the area of a parallelogram (base × height) and dividing by two (since two identical trapeziums make one parallelogram), we derive the formula for the area of a trapezium.

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