Will Approach 2 work for any type of trapezium?

Question

Homi · Accepted Answer

We can find the area of any trapezium by splitting it into a parallelogram and a triangle.

Step 1 — Understanding the trapezium

The diagram shows a shape called a trapezium. A trapezium is a four-sided figure with one pair of parallel sides. In our diagram, side AB is parallel to side DC. The length of side AB is $a$ . The length of side DC is $b$ . The height of the trapezium is $h$ . This is the perpendicular distance between the parallel sides.

Step 2 — The construction in Approach 2

Approach 2 involves drawing a line BG. This line BG is drawn from point B. It is drawn parallel to the side AD. Point G lies on the side DC.

Step 3 — Decomposing the trapezium

This construction divides the trapezium ABCD into two simpler shapes. The first shape is ABGD. In ABGD, we know AB is parallel to DG (because AB is parallel to DC). We also know AD is parallel to BG (by our construction). A four-sided figure with both pairs of opposite sides parallel is a parallelogram. So, ABGD is a parallelogram. The second shape is triangle BGC.

Step 4 — Why this works for any trapezium

The definition of a trapezium only requires one pair of parallel sides. The construction of drawing a line parallel to a non-parallel side (like BG || AD) is always possible for any trapezium. This construction will always create a parallelogram (like ABGD) and a triangle (like BGC). The area of the trapezium is the sum of the areas of this parallelogram and this triangle. The formulas for the area of a parallelogram ( $\text{base} \times \text{height}$ ) and a triangle ( $\frac{1}{2} \times \text{base} \times \text{height}$ ) are general. When we combine these areas, we get the general formula for the area of a trapezium: $\frac{1}{2} \times h \times (a+b)$ . This formula works for all types of trapeziums, whether they are isosceles, right-angled, or scalene. Since the method leads to the general formula, it works for any type of trapezium.

Answer

Yes, Approach 2 will work for any type of trapezium. This method decomposes any trapezium into a parallelogram and a triangle, which allows us to calculate its area using standard formulas.

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