Geometry Concepts: Quadrilaterals

This is the fourth post of Geometry Concept series in which we will learn about the quadrilaterals and parallelograms: rectangle, rhombus and square and trapezium and their properties.


Quadrilateral

A four-sided closed figure is called a quadrilateral. ABCD is quadrilateral.
SSC Maths Study Material

1. Area of a Quadrilateral = ½ X one of the diagonals X sum of perpendiculars drawn to the diagonals from the opposite vertices

So, Area of  ABCD = ½ X AC X (DE + FB)

2. Perimeter of a Quadrilateral = Sum of all sides 

So, Perimeter of ABCD = AB + BC + CD + DA
Properties of a Quadrilateral

1. Sum of four interior angles is 360°. ie., ∠A + ∠B + ∠A + ∠D = 360°.

2. Sum of opposite sides of a quadrilateral circumscribed about a circle is always equal.

3. Figure formed by joining the mid-points of a quadrilateral is parallelogram. 
Parallelogram

A quadrilateral whose opposite side are parallel is known as a parallelogram.
SSC Quantitative Aptitude Study Material
1. Area of a Parallelogram =  Base X Height or  Product of any two sides X sine of the included angle

So, Area of  ABCD = AB X DE.

2. Perimeter  = Sum of all sides
So, Perimeter of ABCD = AB + BC + CD + DA

Properties of a Parallelogram
SSC Maths Study Material
  1. Opposite sides are parallel and equal.
  2. Opposite angles are equal.
  3. Sum of two adjacent angles is 180°.
  4. Diagonals bisect each other.
  5. Each diagonal divided a parallelogram into two congruent triangles.
  6. Lines joining the mid-points of adjacent sides of a quadrilateral is a parallelogram.
  7. Lines joining the mid-points of adjacent sides of a parallelogram is a parallelogram.
  8. Parallelogram that is inscribed in a circle is a rectangle.
  9. Parallelogram that is circumscribed about a circle is rhombus. 
  10. Parallelograms that lie on the same base and between the same parallel lines are equal in area.  
  11. Area of a triangle is half the area of a parallelogram which lie on the same base and between the same parallel lines. 
  12. A parallelogram is rectangle if its diagonals are equal.
Rectangle

A parallelogram in which all the four angles are right angles is a rectangle.

1. Area of a Rectangle = Length X Breadth

2. Perimeter = Sum of all sides 
SSC Quantitative Aptitude Study Material
Properties of a Rectangle
  1. Opposite sides are parallel and equal.
  2. Opposite angles are 90°.
  3. Diagonals are equal and bisect each other.
  4. When a rectangle is inscribed in a circle, the diameter of the circle is equal to the diagonal of the rectangle.
  5. For the given perimeter of rectangle, a square has maximum area.
  6. Figure formed by joining the mid-points of the adjacent sides of a rectangle is a rhombus.
  7. Quadrilateral formed by joining the mid-points of a of intersection of the angle bisectors of parallelogram is a rectangle.
  8. Every rectangle is parallelogram.
Rhombus

A parallelogram in which all the four sides equal.
SSC Maths Study Material

1. Area of a Rhombus = ½ X product of the diagonals

or 

Area of a Rhombus = Product of adjacent sides X sine of the included angle  

Properties of a Rhombus
  1. Opposite sides are parallel and equal.
  2. Opposite angles are equal.
  3. Diagonals bisect each other at right angles.
  4. Diagonals also bisect vertex angles.
  5. Sum of any two adjacent angles is 180°.
  6. Figure formed by joining the mid-points of adjacent sides of a rhombus is rectangle.
  7. A parallelogram is rhombus if its diagonals are perpendicular to each other.
Square

A rectangle whose all sides are equal or a rhombus whose angles are right angles is called a square. 
SSC Maths Study Material
1. Area of a Square = (side)² = a² = (diagonal)² / 2

where diagonal  = side√2

2. Perimeter = Sum of all sides

Properties of a Square
  1. All sides are equal and of course opposite sides are parallel.
  2. All angles are right angles.
  3. Diagonals are equal and bisect each other at right angle. 
  4. Diagonal of an inscribed square is equal to diameter of the inscribing circle.
  5. Side of a circumscribed square is equal to the diameter of the inscribed circle.
  6. The figure formed by joining the mid-points of the adjacent sides of a square is a square.
Trapezium

A quadrilateral whose only one pair of sides is parallel and other two sides are not parallel.  
SSC Maths Study Material
1. Area of a trapezium = ½ X (sum of parallel sides) X height

i.e., Area of ABCD = ½ X (AB + CD) X DM

2. Perimeter of trapezium = sum of all sides

Properties of Trapezium 
  1. The line joining the mid-point of non-parallel sides is half the sum of the parallel sides and is called the median. 
  2. If the non-parallel sides are equal then the diagonals will also be equal to each other.
  3. Diagonals intersect each other proportionally in the ratio of lengths of parallel sides. 
  4. By joining the mid-points of adjacent sides of a trapezium four similar triangle are formed.
  5. If a trapezium is inscribed is a circle then its non-parallel sides are equal.
  6. AC² + BD²  = BC² + AD² + 2 AB.CD
Kite

A kite is a figure having two pairs of adjacent sides are equal.
SSC Quantitative Aptitude Study Material
Properties of Kite
  1. AB = BC and AD = CD
  2. Diagonals intersect at right angles.
  3. Shorter diagonal is bisected longer diagonal.
  4. Area = ½ X product of diagonals

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