# Quadrilaterals

Philosophy of Mathematics |

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### Concept Map

## Textbook

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## Additional Information

This videos is related to classification and properties of quadrilaterals.

### Useful websites

- click here : For effective introduction to quadrilaterals.
- click here : Simple explanation about quadrilaterals.
- click here : This website has a very good activity on properties of quadrilaterals.
- click here This is a very good website for students to understand classification of quadrilaterals as per their properties.

### Reference Books

- Please download 9th standard mathematics textbook of Tamil Nadu state syllabus from the following link and refer the page 89 click here
- Refer 9th standard mathematics ncert textbook from the following link click here

### Additional Resources

#### Resource Title

#### OER

- List web resources with a brief description of what it contains; how it can be used and whether it can be by teacher/ student or both
- Books and journals
- Textbooks
- Syllabus documents

#### Non-OER

- List web resources with a brief description of what it contains; how it can be used and whether it can be by teacher/ student or both
- http://www.mathopenref.com/quadrilateral.html : Simple explanation about quadrilaterals.
- http://www.slideshare.net/muzzu1999/types-of-quadrilaterals-and-its-properties-group-4 : This website has a very good activity on properties of quadrilaterals.
- http://www.cimt.plymouth.ac.uk/projects/mepres/book8/bk8i1/bk8_1i3.htm This is a very good website for students to understand classification of quadrilaterals as per their properties.
- http://www.shodor.org/ihnteractivate/discussions/Quadrilaterals/ click here : For effective introduction to quadrilaterals.

- Books and journals
- Please download 9th standard mathematics textbook of Tamil Nadu state syllabus from the following link and refer to page 89 click here
- Refer 9th standard mathematics NCERT textbook from the following link click here

- Textbooks : Karnataka State Text book of mathematics Class 9-Chapter 8:Quadrilaterals
- Syllabus documents (CBSE, ICSE, IGCSE etc)

# Additional Information

An Ortho-diagonal quadrilateral i.e., any quadrilateral whose diagonals are perpendicular to each other possesses certain interesting properties. This article 'Quadrilaterals with Perpendicular Diagonals' by Shailesh Shirali (published in *'At Right Angles'* | Vol. 6, No. 2, August 2017) discusses a few of them.

### Learning Objectives

- Introduction to polygons
- The meaning of quadrilateral
- Identification of various types of quadrilaterals
- Different properties of special quadrilaterals
- Construction of quadrilaterals to given suitable data
- Finding area of quadrilaterals
- Introduction to cyclic quadrilaterals

### Teaching Outlines

#### Concept 1: Introduction to Quadrilaterals

The word quadrilateral comes from two latin words "quadri" which means a "variant of 4" and 'latera' which means 'side'. A quadrilateral is a 4 sided figure with 4 sides, 4 angles and 4 vertices.

This topic has its basics in polygons. Try to elicit live examples for quadrilaterals from within classroom, starting from the shape of a textbook. Show the vertices of a rectangular page. Mark three sets of four points on the blackboard, one set being collinear and other non-collinear. Call students to join the points of each set of points. This activity will introduce them to the concept of quadrilateral.

##### Activities #

###### Introduction to quadrilaterals

This activity explores formation of a quadrilateral and elements related with the shape.

###### Identifying quadrilaterals

This is an exploration into quadrilaterals. A specific type of quadrilateral can be selected with the checkboxes, and any blue dots on each quadrilateral can be dragged to change the shape.

###### Concept 3: Types of quadrilaterals

Quadrilaterals are of different types. Grouping is made based on the four angle measures and/or sides. Each type is recognized with its characteristic properties. The types include regular, non-regular; convex, concave; parallelogram (square, rectangle, rhombus,) and non-parallelograms (trapezium and kite).

##### Activities #

###### "I have - Who has ?"

A hands on group activity that helps in identifying and building vocabulary related to quadrilaterals.

###### Venn diagrams of quadrilaterals

Classifying quadrilaterals based on their properties and identifying related quadrilaterals with ven-diagram.

#### Concept 2: Properties of quadrilaterals

There are certain characteristic properties by which a quadrilateral is identified. A quadrilateral is a plane closed figure having 4 sides and 4 angles. The sum of all 4 interior angles of any quadrilateral always equals to 360 degrees. This is called the interior angle sum property of a quadrilateral. The sum of all 4 exterior angles of any quadrilateral equals 360 degrees. This is called the exterior angle sum property of the quadrilateral. If any 3 angles of a quadrilateral are known the fourth angle can be found using the angle sum property.

##### Activities #

###### Angle sum property of a quadrilateral

Showing the sum of angles of quadrilaterals by placing the angles of the quadrilateral adjacent to each other with a hand on activity.

###### Sum of the interior angles of a quadrilateral

The sum of the measures of the angles in any quadrilateral is 4 right angles.

###### Sum of angles at point of intersection of diagonals in a quadrilateral

A diagonal is the line segment that joins a vertex of a polygon to any of its non-adjacent vertices. These two diagonals of a quadrilateral form angle, this activity explores the property of these angles.

###### Area of a quadrilateral

A diagonal divides a quadrilateral into 2 triangles. Understanding the area of a quadrilateral in terms of triangles is done with this activity.

#### Properties of Parallelogram

#### Parallelogram on same base and between same parallels have equal area

#### Mid point of sides of a Quadrilateral forms parallelogram

#### Concept 3 : Properties of Rhombus

A rhombus is a quadrilateral with all sides of equal length. The opposite angles of a rhombus are equal and its diagonals are perpendicular bisectors of one another. Since the opposite sides and opposite angles of a rhombus have the same measures, it is also a parallelogram. Hence, a rhombus has all properties of a parallelogram and also that of a kite. (click here)

#### Concept 3: Construction of quadrilaterals

#### Concept 4: Square

A square is a 4-sided regular polygon with all sides equal and all internal angles 90. A square is the only regular quadrilateral. It can also be considered as a special rectangle with both adjacent sides equal. Its opposite sides are parallel. The diagonals are congruent and bisect each other at right angles. The diagonals bisect the opposite angles. Each diagonal divides the square into two congruent isosceles right angled triangles. A square can be inscribed in a circle. A circle can be inscribed in a square touching all its sides.

Click here : Introduction to a square and its properties

Four sides of a square are equal. Adjacent sides are at right angles with each other. The area of a square is side x side sq units. The perimeter of a square is the length of distance around its boundary which is 4 times its side.

###### Pull me to see if I still remain a square

###### Area of a square

#### Concept 5: Cyclic Quadrilaterals

#### Cyclic Quadrilaterals

###### Theorems on cyclic quadrilaterals

#### Concept : Kite

A kite has two pairs of congruent sides. Its diagonals intersect at right angles. The sum of its four sides would be its perimeter. Its area is given by the formula

###### A Kite and its properties

###### Construction of a kite

###### Deriving formula for area of a kite

#### Concept : Trapezium

###### A Trapezium and its properties

###### Deriving formula for area of a trapezium

###### Construction of Trapezium

###### Construct an isosceles trapezium and study its properties

A trapezium in which non-parallel sides are equal is called as an Isosceles Trapezium. The diagonals of an isosceles trapezium are equal. An isosceles trapezium has one line of reflection symmetry. This line connects the midpoints of the two bases. Both pairs of base angles of an isosceles trapezium are congruent. Pairs of angles in an isosceles trapezium that do not share a base are supplementary. The area of an isosceles trapezium is given by , where a and b are the lengths of the parallel sides and h is the distance (height) between the parallel sides.