Quadrilateral Basics

What is a Quadrilateral?

Quadrilateral is a two-dimensional enclosed figure formed by joining four points on a plane. Geometry

So, a quadrilateral has four vertices and four sides.

Some examples of quadrilaterals are: Parallelogram, Trapezium, Kite etc.

Types of Quadrilaterals

Parallelogram

It is a quadrilateral with two pairs of parallel sides. Geometry

  • The opposite sides of a parallelogram are parallel and are of equal length.
  • The opposite angles of a parallelogram are also equal. That is, ∠ A = ∠ C; ∠ B = ∠ D

There are various kinds of parallelograms, such as Rhombus, Rectangle, Square. We will study about these in much more detail in a separate article.

Trapezium

It is a quadrilateral with one pair of parallel sides. Geometry

In the above figure, AB ∥ CD

Isosceles trapezium - A trapezium whose non-parallel sides are equal.
That is, AB ∥ CD and BC = AD

Kite

It is a quadrilateral in which two pairs of adjacent sides are equal. Geometry

In the above figure, AB = AD, and CB = CD

  • A Kite is not a parallelogram.

  • Square and Rhombus are both Parallelogram and Kite.

Cyclic Quadrilateral

Cyclic Quadrilateral is a kind of quadrilateral, all of whose vertices lie on the circumference of a circle. Geometry

Circumscribed Quadrilateral

Circumscribed Quadrilateral is a convex quadrilateral, all of whose four sides are tangent to a single circle within it. That’s why it’s also called Tangential Quadrilateral. Geometry




Properties of Quadrilaterals

Let’s see some of the properties shared by all types of quadrilaterals.

Property 1: Angle Sum

Sum of the four interior angles of a quadrilateral is 360°. Geometry In the above quadrilateral, ∠ A + ∠ B + ∠ C + ∠ D = 360°

Similarly, Sum of the four exterior angles of a quadrilateral is also 360°.

Property 2

If we join the mid-points of the four sides of a quadrilateral, then we will get a parallelogram. The area of this parallelogram will be half of the original quadrilateral. Geometry

In the above figure, on joining the mid-points of the sides of the quadrilateral □ABCD, we get a parallelogram PQRS.

Area of parallelogram PQRS = $\frac{1}{2}$ × Area of □ABCD

If we join the mid-points of the four sides of a parallelogram, then we will get a parallelogram. The area of this parallelogram will be half of the original parallelogram.

This is obvious, as a parallelogram is also a quadrilateral.

Property 3

Angle made by the bisectors of any two consecutive angles = Half of the sum of the other two angles Geometry

In the above figure, ∠AOB = $\frac{1}{2}$ × (∠C + ∠D)

Property 4

Sum of a pair of interior opposite angles = Sum of the pair of other two exterior opposite angles Geometry

In the above figure, ∠x° + ∠y° = ∠p° + ∠q°

Property 5

The line segment joining midpoints of any two adjacent sides is parallel and equal to half of corresponding diagonal. Geometry

In the above figure, PQ ∥ AC and PQ = $\frac{1}{2}$ × AC

It is an application of the Thales theorem (or Basic proportionality theorem) that we read in triangle module.

Property 6

Perimeter of a quadrilateral is always greater than the sum of its diagonals. Geometry

In the above figure, AB + BC + CD + DA > AC + BD

Properties of Kites

Property 1: Diagonals

Diagonals of a Kite intersect at 90° and the shorter diagonal is bisected by the longer diagonal. Geometry

Properties of Cyclic Quadrilaterals

Property 1: Sum of Opposite angles

In any cyclic quadrilateral, the sum of either pair of opposite angles = 180°. Geometry

That is, ∠A + ∠C = 180°
And, ∠B + ∠D = 180°

Property 2

If the two diagonals of a cyclic quadrilateral ABCD intersect each other at a point P, then: Geometry

AP × PC = DP × PB

Property 3: In case of Square

Diagonal of inscribed square = Diameter of circle Geometry

Properties of Circumscribed Quadrilaterals

Property 1

Sum of opposite sides of a Circumscribed Quadrilateral are equal. Geometry

In the above figure, AB + CD = BC + AD

Property 2: In case of Square

Side of circumscribed square = Diameter of the inscribed circle Geometry

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