Quadrilaterals
Quadrilaterals Synopsis
Synopsis
- Introduction to quadrilateral.
A quadrilateral is a closed figure obtained by joining four points (with no three points collinear) in an order.
Here, ABCD is a quadrilateral.
Parts of a quadrilateral
- A quadrilateral has four sides, four angles and four vertices.
- Two sides of a quadrilateral having no common end point are called its opposite sides.
- Two sides of a quadrilateral having a common end point are called its adjacent sides.
- Two angles of a quadrilateral having common arm are called its adjacent angles.
- Two angles of a quadrilateral not having a common arm are called its opposite angles.
- A diagonal is a line segment obtained on joining the opposite vertices.
- Angle sum property of a quadrilateral
Sum of all the angles of a quadrilateral is 360º. This is known as the angle sum property of a quadrilateral.
- Types of quadrilaterals and their properties:
Important facts about quadrilaterals
- If the non-parallel sides of trapezium are equal, it is known as isosceles trapezium.
- Square, rectangle and rhombus are all parallelograms.
- Kite and trapezium are not parallelograms.
- A square is a rectangle.
- A square is a rhombus.
- A parallelogram is a trapezium.
Formation of a new quadrilateral using the given data
- If the diagonals of a parallelogram are equal, then it is a rectangle.
- If the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.
- If the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
If there are three or more parallel lines and the intercepts made by them on a transversal are equal, then the corresponding intercepts on any other transversal are also equal.
- Area of Polygons
- Area of a Quadrilateral
Area of a quadrilateral × One of the diagonals × Sum of perpendiculars drawn on the diagonals from opposite vertices.
Area of a Parallelogram
- Area of a parallelogram = base × height
- Area of a parallelogram = Product of any two adjacent sides × sine of included angle
- Perimeter of a parallelogram = 2 × Sum of any two adjacent sides
- Parallelograms that lie on the same base and between the same parallel lines are equal in area.
Area of a rectangle
- Area of a rectangle = length × breadth = l × b
Area of a Rhombus
- Area of Rhombus = base × height
- Area of Rhombus = Product of adjacent sides × sine of included angle
Area of a Square
- Perimeter of a Square =4× side
Area of a Trapezium
Area of a Kite
Cyclic Quadrilateral
- A quadrilateral whose all four vertices lie on a circle is called cyclic quadrilateral.
- The sum of pair of opposite angles of a cyclic quadrilateral is 180o.
- If a side of a cyclic quadrilateral is produced, then the exterior angle so formed is equal to the interior opposite angle.
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