Properties of Quadrilaterals

A quadrilateral is a special type of polygon that has exactly four sides, four vertices, and four angles. The word "quadrilateral" is derived from the Latin words "quadri" meaning four and "latus" meaning side. In Class 8 Maths, we delve deeper into the properties of these interesting shapes, their types, and how to identify them using their properties.

Basic properties

Before we explore specific types of quadrilaterals, let's look at the basic properties that all quadrilaterals share:

• They all have four arms.
• The sum of the interior angles of any quadrilateral is always 360 degrees.
• They all have two diagonals.

Interior angle

The formula for calculating the sum of the interior angles of any polygon is (n-2) * 180 degrees, where n is the number of sides. For a quadrilateral, n is 4, and this leads to:

(4-2)*180 = 360 degrees

This means that no matter what the shape of the quadrilateral, if you add up all its interior angles, it will always equal 360 degrees. For example:

• If one angle is 90 degrees, the second is 80 degrees, the third is 120 degrees, then the fourth angle will be 360 - (90 + 80 + 120) = 70 degrees.

Diagonals

The diagonals of a quadrilateral are the lines connecting opposite vertices. Every quadrilateral has two diagonals. They help in dividing the quadrilateral into smaller triangles and also play an important role in defining the properties of specific quadrilaterals like parallelogram, rhombus, etc.

Types of quadrilaterals

Quadrilaterals are classified into different types based on their characteristics such as sides, angles and diagonals:

Quadrilateral

A parallelogram is a quadrilateral that has two pairs of parallel sides. The opposite sides are also equal in length.

• Opposite angles are equal.
• The sum of consecutive angles is 180 degrees.
• The diagonals bisect each other.

Example:

• If one angle of a parallelogram is 100 degrees, then the opposite angle will also be 100 degrees. Adjacent angles will be 180 - 100 = 80 degrees each.

Rectangle

A rectangle is a special type of parallelogram, all of whose angles are right angles (90 degrees).

• The opposite sides are equal and parallel.
• The diagonals are equal and bisect each other.

Example:

• The diagonals of a rectangle not only bisect each other but are also equal. If the measure of one diagonal is 5 cm, then the other is also 5 cm.

Rhombus

A rhombus is a parallelogram in which all sides are equal in length.

• Opposite angles are equal.
• The diagonals bisect each other at right angles (90 degrees).
• The diagonals bisect the angles of a rhombus.

Example:

• In a rhombus, if one angle is 60 degrees, then the opposite angle will also be 60 degrees, and the adjacent angles will be 180 - 60 = 120 degrees each.

Square

A square is a special type of rectangle in which all sides are of equal length.

• All angles are right angles.
• The diagonals are equal and bisect each other at right angles.

Example:

• The diagonal of a square divides it into two equal right-angled triangles.

Trapezoid (or trapezium)

A trapezoid is a quadrilateral with at least one pair of parallel sides.

• If it has two pairs of parallel sides, it becomes a parallelogram.
• The parallel sides are called bases.
• The other two sides are non-parallel and are called the legs of the trapezium.

Example:

• In an isosceles trapezoid, the lengths of the legs are equal, and the angles at the base are also equal.

Kite

A kite is a quadrilateral with two different pairs of equal adjacent sides.

• The diagonals are perpendicular to each other.
• One diagonal bisects the other.
• The longer diagonal bisects the angles of the kite.

Example:

• In a kite, if the longer diagonal is 8 cm and it bisects the other diagonal of 6 cm, then the kite is divided into four triangles: two equilateral and two right-angled triangles.

Identifying quadrilaterals

To identify a quadrilateral it is important to have a clear understanding of its properties. Here are some simple steps that can be followed:

• Check the number of parallel sides present. This may help in classifying the quadrilateral as a trapezoid or parallelogram etc.
• Verify the measurement of angles: All angles in rectangles and squares will be right angles.
• Measure the lengths of the sides and diagonals. For example, in a rhombus, all four sides are equal and in a rectangle, the diagonals are equal.
• Look at the properties of diagonals - if they bisect at right angles, it indicates a rhombus or square.

Formulas related to quadrilaterals

In problems involving quadrilaterals, understanding and using formulas can help solve them more easily. Here are some important formulas:

Area

• For a rectangle, the area is calculated as: Area = length × width
• For a square, the area is: Area = side²
• For parallelogram: Area = base × height
• For trapezium: Area = ½ × (base₁ + base₂) × height

Circumference

• For a rectangle, the perimeter is: Perimeter = 2 × (length + width)
• For a square, the perimeter is: Perimeter = 4 × side
• For any quadrilateral: Perimeter = sum of all sides

Real-life applications

Quadrilaterals are not just confined to geometry textbooks; they have many applications in real life too. Here are some examples:

• Architecture: Most buildings and rooms are designed using rectangles and squares because of their stability and practicality in construction.
• Tiling: Tiles often come in square shapes, so that floors and walls can be covered efficiently without unnecessary gaps.
• Design: Parallelograms and trapezoids can often be seen in sewing and framework designs.

Conclusion

Understanding the properties of quadrilaterals provides a strong foundation in geometry. Learning to identify the types based on sides, angles, and diagonals is important not only in solving mathematical problems but also in recognizing their applications in real-world scenarios. From parallelograms that serve as the basis for further shapes, to trapezoids with their unique applications, each quadrilateral has its own significance in both theory and practice.

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