**Midsegment Theorem on Trapezium:** Initially, you should be aware of the trapezoids as they help you to perceive the concept of a midsegment theorem of trapezoids from a different point of view. In geometry, the meaning of trapezoids is a 2D figure with a four-sided polygon that covers some area and also has a perimeter.

A trapezoid or trapezium has a set of opposite sides that are parallel and a set of non-parallel sides. Let us discuss more regarding the concept of a trapezium and prove the Midsegment Theorem statement. We will have a glimpse of the properties of trapezoids and the formulas used to solve a trapezoid problem.

Do Refer:

## Trapezoids – Definition

Trapezoids are the type of quadrilaterals that have one pair of parallel sides and it is also known as a trapezium in some areas. The midsegment of a trapezoid is the segment connecting the midpoints of the two non-parallel sides. In a trapezium the parallel sides are bases and the non-parallel sides are legs. It is a 2D shape similar to a square, rectangle, and parallelogram. This trapezoid also has an area and a perimeter as other shapes have.

### Area of a Trapezoid

The area of a trapezoid is defined as the area occupied by the trapezoid. The area is calculated by the average of two bases and product it with the altitude. The area of a trapezoid is given as follows:

Area of a trapezoid = ½(a+b)h

where a= shorter base

b= longer base

h= height or distance between the two bases (altitude).

### The Perimeter of a Trapezoid

The perimeter of a trapezoid is defined as the sum of all its sides. Consider a trapezoid ABCD as shown below with sides a, b, c, and d. The formula for the perimeter can be written as

Perimeter of a trapezoid, P = Sum of all its sides = a+ b+ c+ d

where a, b, c, and d are sides of the trapezoid.

### Properties of Trapezoid Midsegment

There are certain Trapezoid Midsegment properties and they are as follows:

- A midsegment of a trapezoid is a segment that connects the midpoints of the two non-parallel sides of a trapezoid.
- A trapezoid midsegment is parallel to the set of parallel lines in a trapezoid and is equal to the average of the lengths of the bases.
- A triangle midsegment is related to a midsegment trapezoid, given that both of their lengths are proportional to the bases.

### Midsegment Theorem on Trapezoid with Proof

Prove that the line segment joining the midpoints of the non-parallel sides of a trapezium is half the sum of the lengths of the parallel sides and is also parallel to them.

#### Given:

ABCD is a trapezium in which AB ∥ CD. E and F are the midpoints of BC and AD respectively.

#### To Prove:

Consider the above trapezoid ABCD and prove that

EF ∥ CD and EF = ½(AB + CD).

#### Construction:

Construct a line through the points BF and produce it to meet CD produced at the point G.

#### Trapezoid Midsegment Theorem Proof:

In ∆ABF and ∆DGF

(i) AF = FD (by given)

(ii) ∠AFB = ∠ GDF (Vertically opposite angles)

(iii) ∠BAF = ∠FDG (Alternate angles)

Therefore, ∆ABF ≅ ∆DGF (by ASA theorem of congruence)

Thus, AB = DG (by c.p.c.t.c)

BF = FG (by c.p.c.t.c)

In ∆BCG,

(i) E is the midpoint of BC (by given)

(ii) F is the midpoint of BG (since BF =FG)

Therefore, EF ∥ CG and EF = ½CG (by the midpoint theorem)

⇒ EF = ½(CD + DG)

⇒ EF = ½(CD + AB) (since DG = AB)

Therefore, EF ∥ CD and EF = ½(CD + AB) ( from the above statements EF ∥ CG and EF = ½(CD + AB) ).

Hence, the given statement is proved.

### FAQ’s on Midsegment of Trapezoid

**1. What is the midsegment theorem of a trapezoid? **

The trapezoid of a midsegment theorem states that the length of the midsegment is equal to the sum of the base lengths divided by two. It is also defined as the midsegment is the average length of the two bases.

**2. How do you prove the trapezoid Midsegment Theorem?**

To prove the theorem, it has to satisfy the two properties when the midpoints of any two sides of a triangle are connected with a line segment. Firstly, the line segment should be parallel to the third side and the other, the length of the line segment will be half the length of the third side.

**3. What does a Midsegment mean?**

Each side of the medial triangle is called a midsegment/ a midline. In other words, a line joining the midpoints of two sides of the triangle is called a midsegment.

**4. Which condition makes a quadrilateral a trapezium?**

A quadrilateral is a trapezium if it has only one pair of parallel sides. The parallel sides are termed as **bases** and the other sides/ non-parallel sides are termed as **legs** or lateral sides.

**5. How do you find the trapezoid Midsegment theorem?**