1. Definition of a Trapezium
A trapezium is a quadrilateral that has at least one pair of parallel sides. These parallel sides are called the bases of the trapezium.
If the trapezium is named as \(ABCD\) and \(AB \parallel CD\), then \(AB\) and \(CD\) are its bases, and the other two sides are called non-parallel sides or legs.
2. Parts of a Trapezium
Understanding its structure makes it easier to remember formulas and solve problems.
2.1. Bases
The two parallel sides of a trapezium are called bases. Usually, one base is longer than the other.
2.2. Legs or Non-Parallel Sides
The remaining two sides, which are not parallel, are called the legs of the trapezium.
2.3. Height
The height is the perpendicular distance between the two bases. It is very important for area calculation.
2.4. Diagonals
Like any quadrilateral, a trapezium has two diagonals: \(AC\) and \(BD\). They are not usually equal or perpendicular.
3. Special Types of Trapezium
Depending on the side lengths and angle conditions, trapeziums can have special forms.
3.1. Isosceles Trapezium
If the non-parallel sides (legs) of a trapezium are equal in length, it is called an isosceles trapezium.
- Base angles are equal.
- Diagonals are equal.
3.2. Right Trapezium
If one of the legs is perpendicular to the bases, the trapezium is called a right trapezium.
It has exactly two right angles.
4. Mid-Segment (Median) of a Trapezium
The line segment joining the midpoints of the two non-parallel sides of a trapezium is called the mid-segment or median.
4.1. Properties of the Mid-Segment
- The mid-segment is parallel to both bases.
- Its length is the average of the lengths of the bases.
4.1.1. Length Formula
\(\text{Mid-segment} = \dfrac{1}{2}(a + b)\)
where \(a\) and \(b\) are the lengths of the two bases.
5. Area of a Trapezium
The area depends on the lengths of the two bases and the height.
5.1. Area Formula
If bases are \(a\) and \(b\), and height is \(h\), then:
\(\text{Area} = \dfrac{1}{2}(a + b)h\)
5.2. Quick Example
For a trapezium with bases \(8\text{ cm}\) and \(12\text{ cm}\), and height \(5\text{ cm}\):
\(\text{Area} = \dfrac{1}{2}(8 + 12) \times 5 = \dfrac{1}{2}(20) \times 5 = 50\text{ cm}^2\)
6. Trapezium in Geometry
Trapeziums often appear in coordinate geometry, mensuration, and proof-based problems because of their mix of parallel and non-parallel sides.
The mid-segment property and area formula are the most commonly used ideas.