Question 38 of 42advanced🔍 AnalyzeLong Answer5 marks

By dividing a trapezium into two triangles show that its area is, half the sum of the parallel sides multiplied by the height (the same formula as the one given above).

Correct Answer

Let us consider a trapezium with parallel sides of length aa and bb, and a height hh. We can divide this trapezium into two triangles by drawing one of its diagonals.

The diagonal divides the trapezium such that one triangle has the side aa as its base. Since the height of this triangle is the same as the height of the trapezium, its area is 12×a×h\frac{1}{2} \times a \times h, which equals ah2\frac{ah}{2}.

The second triangle formed by the diagonal has the side bb as its base. This triangle also shares the same height hh as the trapezium. Therefore, the area of this triangle is 12×b×h\frac{1}{2} \times b \times h, which equals bh2\frac{bh}{2}.

The area of the trapezium is the sum of the areas of these two triangles. So,

Area of trapezium=ah2+bh2\text{Area of trapezium} = \frac{ah}{2} + \frac{bh}{2}

Taking h2\frac{h}{2} as the common factor, we get:

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

Thus, we have shown that the area of a trapezium is half the sum of the parallel sides multiplied by the height.

Exercise: END-OF-CHAPTER EXERCISES | Q: 12 | (Page 33)
For More Understanding

Explanation

The question requires deriving the area of a trapezium by splitting it into two triangles. The context provides the formula for the area of a triangle as 12×base×height\frac{1}{2} \times \text{base} \times \text{height}. By drawing a diagonal across a trapezium with parallel sides aa and bb and height hh, it is divided into two triangles. One triangle has base aa and height hh, while the other has base bb and height hh. Adding their individual areas algebraically leads directly to the standard formula for the area of a trapezium.

Solution Steps

  1. Step 1: Consider a trapezium with parallel sides aa and bb, and height hh. Draw a diagonal to divide it into two triangles.

  2. Step 2: Calculate the area of the first triangle with base aa and height hh:

    Area1=12×a×h=ah2\text{Area}_1 = \frac{1}{2} \times a \times h = \frac{ah}{2}

  3. Step 3: Calculate the area of the second triangle with base bb and height hh:

    Area2=12×b×h=bh2\text{Area}_2 = \frac{1}{2} \times b \times h = \frac{bh}{2}

  4. Step 4: Add the areas of the two triangles:

    Area of trapezium=ah2+bh2\text{Area of trapezium} = \frac{ah}{2} + \frac{bh}{2}

  5. Step 5: Factor out h2\frac{h}{2} to get the final formula:

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