The Parallel Sides of a Trapezium

Introduction to Trapezium Areas

A trapezium is a quadrilateral with one pair of parallel sides. If we consider a trapezium with parallel sides of 30 cm and 20 cm, and non-parallel sides of 13 cm each, we can determine the area using geometric principles and algebra. The goal is to find the distance between the parallel sides, which is the height of the trapezium.

Calculation Method

Let's proceed by breaking down the steps:

Identify the given sides: Let AB 30 cm, CD 20 cm, BC 13 cm, AD 13 cm. Construct a perpendicular (CP) from point C to AB: Since CP is perpendicular to AB, CP AD 13 cm. Find the length of BQ: BQ AB - AQ 30 - 20 10 cm. Find the height (h) of the trapezium: The area of triangle CQB is 1/2 * 13 * 15 * 10 * (19 - sqrt(19*19 - 15*15 - 13*13 10*10)) / (19*19 - 15*19 - 13*19 10*19) 19 * 4 * 6 * 9 6114 The area of triangle CQB is also 1/2 * CP * BQ 1/2 * 13 * 10 12114 / 10 12 cm. Calculate the area of the trapezium: Area 1/2 * (AB CD) * CP 1/2 * (30 20) * 12 152 cm2.

The area of the trapezium can be calculated as: 320.31 sq.units.

Case Study: Another Trapezium Example

Let's consider another example, where ABCD is an isosceles trapezium in which AB 28 cm, CD 14 cm, AD BC 15 cm. We draw CP parallel to AD meeting AB at P. Here are the detailed steps:

Identify the base lengths and heights: Find the height of the trapezium: AP PQ CD 14 cm. BP AB - AP 28 - 14 14 cm. The height of the trapezium is the altitude of triangle PBC. Using Heron's formula, the area of triangle PBC is calculated as: s (13 15 14) / 2 21, area sqrt(21 * (21 - 13) * (21 - 15) * (21 - 14)) 84. The height (CM) of the trapezium is 12 cm. Calculate the area of the trapezium: Area 1/2 * (28 14) * 12 252 cm2.

Conclusion

Understanding and applying the principles of geometry, particularly in the calculation of trapezium areas, can help in solving various problems. Whether it's through direct algebraic methods or geometric constructions, the key is to break down the trapezium into manageable parts and use geometric properties to find the area.