Calculating the Length of Non-Parallel Sides of a Trapezium: A Comprehensive Guide

In this article, we delve into the method of calculating the length of the non-parallel sides of a trapezium, focusing on the properties of isosceles trapezia and the application of geometric principles. We will explore several examples to understand the steps involved in such calculations.

Introduction to Trapezium and Isosceles Trapezium

A trapezium is a quadrilateral with at least one pair of parallel sides, known as the bases. When the non-parallel sides of the trapezium are equal in length, it is referred to as an isosceles trapezium. The concept of calculating the length of these non-parallel sides is crucial for solving various geometric problems.

Example 1: Calculating Non-Parallel Sides Given the Area

Given:
Area of the trapezium 168 square meters
Base (longer side) 36 cm
Base (shorter side) 20 cm
The non-parallel sides are equal (isosceles trapezium)

Step 1: Using the Area Formula
Area (T) (AB CD) * h / 2
168 (36 20) * h / 2
168 56 * h / 2
168 28 * h
h 168 / 28
h 6 cm (Height of the trapezium)

Step 2: Using Pythagoras Theorem
Let AF and CE be the heights from the bases AB and CD to the non-parallel sides, forming right triangles ADF and BCE.
In triangle ADF (or BCE), the height is 6 cm and the base is (36 - 20) / 2 8 cm.
Length of the non-parallel side (p) √(AF2 82)
Length of the non-parallel side (p) √(62 82)
Length of the non-parallel side (p) √(36 64)
Length of the non-parallel side (p) √100
Length of the non-parallel side (p) 10 cm

Conclusion: The length of the non-parallel sides of the trapezium is 10 cm.

Example 2: Solving for Non-Parallel Sides Using Another Method

Given:
ABCD is a trapezium with AB CD 10 cm, and AD BC x (non-parallel sides).
Ce is drawn parallel to AD and meets AB at E.
CE x

Step 1: Identifying Congruent Triangles
Since AD BC and CE AB CD, triangle ADE and BCE are congruent and right-angled triangles with height 5 cm and base 10 cm.
EP PB 10 / 2 5 cm
In triangle CEB, x2 52 h2
h √(x2 - 52)
h √(x2 - 25)

Step 2: Using the Area Formula Again
Area of trapezium ABCD (AB CD) * h / 2 (CE * BE)/2
168 (10 20) * h / 2
168 30 * h / 2
168 15 * h
h 168 / 15
h 11.2 cm (Approximately)

Solving for x
x2 - 25 11.22
x2 - 25 125.44
x2 150.44
x √150.44
x ≈ 12.3 cm (Approximately)

Conclusion: The length of the non-parallel sides of the trapezium is approximately 12.3 cm.

Conclusion

Calculating the length of the non-parallel sides of a trapezium, especially an isosceles trapezium, involves understanding the geometric properties and applying the area formula along with the Pythagorean theorem. This guide provides a step-by-step approach to solving such problems, ensuring precision and accuracy in geometric calculations.