Mahesh has a toy which has the shape of a trapezium. The two sides which are parallel have a length of 60 cm and 77 cm and the other sides are 25 cm and 26 cm. He requests you to calculate the areaand you help him to arrive at the correct answer which is sq.cm. [Note:- DO NOT include spaces in your answer)
To calculate the area of a trapezium, we use the formula: \[ \text{Area} = \frac{1}{2} \times (a + b) \times h \] where \(a\) and \(b\) are the lengths of the parallel sides, and \(h\) is the height. Given: - \(a = 60 \text{ cm}\) - \(b = 77 \text{ cm}\) - Other sides are 25 cm and 26 cm. To find the height \(h\), we use the fact that the non-parallel sides and the height form two right-angled triangles when dropped perpendiculars from the endpoints of one parallel side to the other. Using Pythagoras' theorem for these triangles: 1. Split the trapezium into two right-angled triangles by dropping perpendiculars from the ends of the shorter parallel side (60 cm) to the longer one (77 cm), creating a rectangular middle section of width 60 cm and two right triangles with the bases \(\frac{77 - 60}{2} = 8.5 \text{ cm}\). 2. Apply Pythagoras' theorem to find the height \(h\): \[ \sqrt{25^2 - 8.5^2} = \sqrt{625 - 72.25} = \sqrt{552.75} \approx 23.5 \text{ cm}\] or \[ \sqrt{26^2 - 8.5^2} = \sqrt{676 - 72.25} = \sqrt{603.75} \approx 24.5 \text{ cm}\] 3. Approximate average height \(h = (23.5 + 24.5)/2 \approx 24 \text{ cm}\) Using the height to find the area: \[ \text{Area} = \frac{1}{2} \times (60 + 77) \times 24 = \frac{1}{2} \times 137 \times 24 = 68.5 \times 24 = 1644 \text{ sq.cm} \] Thus, the area of the trapezium is 1644 sq.cm.
