Concept: Quandrilaterals & Polygons

 

GENERAL PROPERTIES OF QUADRILATERALS

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  • Sum of all the angles of a quadrilateral is 360°

  • Sum of all the exterior angles of a quadrilateral is 360°

  • Area = ½ × (one diagonal) × (Sum of heights on this diagonal)

  • Area = ½ × d1d2 × 𝑆𝑖𝑛𝜃

  • Sum of the 4 sides > Sum of the 2 diagonals

 

PARALLELOGRAM

General Properties of Parallelograms

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  • Opposite sides are parallel and equal

  • Opposite angles are equal

  • Area = base × height

  • Diagonals bisect each other

  • Adjacent angles are supplementary

  • Each diagonal divides the ||gm in two congruent triangles.

  • AC2 + BD2 = AB2 + BC2 + CD2 + DA2

  • Parallelogram with same base and between same parallel lines will have same area

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RECTANGLE

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  • All properties of a ||gm.

  • All angles are equal to 90°.

  • Both diagonals are equal and bisect each other.

    • AC = BD = l2+b2
    • OA = OB = OC = OD
  • If P is a point inside the rectangle, then

  • PA2 + PC2 = PB2 + PD2

  • Area = l × b

  • Figure formed by joining the midpoints is a rhombus.


SQUARE

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  • All properties of a rectangle.

  • Both diagonals are equal (a√2) and bisect each other at 90°.

  • Diagonals bisect the vertex angles.

  • Area = a2

  • Figure formed by joining the midpoints is a square.


RHOMBUS

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  • All properties of a ||gm.

  • Diagonals are not necessarily equal but bisect each other at 90°

  • Diagonals bisect the vertex angles.

  • d12 + d22 = 4a2

  • Area = ½ × d1d2

  • Figure formed by joining the midpoints is a rectangle.

 

OTHER QUADRILATERALS

KITE

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  • Kite is not a parallelogram.

  • 2 pair of Adjacent sides are equal.

  • Here, AD = AB and CD = CB
  • Bigger diagonal bisects the smaller diagonal at 90°.

  • Area = ½ × d1d2

  • Opposite angles along the smaller diagonal are equal.

  • Here, ∠D = ∠B


TRAPEZIUM

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  • One pair of opposite sides (bases) is parallel.

  • Area = ½ × ℎ𝑒𝑖𝑔ℎ𝑡 × (𝑠𝑢𝑚 𝑜𝑓 𝑏𝑎𝑠𝑒𝑠)

  • The length of a line dividing the oblique sides in the ratio a : b is given by

  • aa + b × CD + ba + b × AB

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  • Diagonals divide the median in the ratio of length of parallel sides

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Isosceles Trapezium

  • If oblique sides are equal, it is called an isosceles trapezium.

  • AD = BC and ∠D = ∠C

  • If a trapezium is inscribed inside a circle, it will always be an isosceles trapezium, i.e. its non−parallel sides will be equal


CYCLIC QUADRILATERAL

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  • A quadrilateral whose all 4 vertices lie on the same circle is called a cyclic quadrilateral. The sum of any two opposite angles of a cyclic quadrilateral is 180°.

  • Area of a cyclic quadrilateral = (s - a)(s - b)(s - c)(s - d)

  • where s = semiperimeter = a + b + c + d2

 

GENERAL PROPERTIES OF POLYGONS

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  • A polygon is a closed figure bounded by three or more straight lines, known as sides. It has as many vertices as the number of sides, with no three of them collinear.

  • A line joining two non−adjacent vertices is known as the diagonal.

  • No. of diagonals in a ‘n’ sided polygon: n(n - 3)2

  • Sum of internal angles of a ′n′ sided polygon: (n - 2)π

  • Sum of exterior angles of a ′n′ sided polygon: 2π

  • A convex polygon has all angles < 180°.

  • A concave polygon has at least one angle > 180°.


REGULAR POLYGONS
  • Measure of each interior angle: n(n - 2)π2

  • Measure of each exterior angle: n

  • Area of a regular ′n′ sided polygon: n × (area of smaller triangle)

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    i.e., n × a24Cotπn

  • Inradius of a n sided regular polygon = a2cotπn

  • Circumradius of a n sided regular polygon: a2cosecπn

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REGULAR HEXAGON (Side = a)
  • Height of a regular hexagon: a√3

  • Distance between opposite vertices: 2a

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  • Area of a regular hexagon: 6 × 3a24

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SOME IMPORTANT RESULTS
  • For a fixed perimeter, the area of a polygon with higher number of sides will always be more than the area of a polygon with lesser number of sides.
  • For any fixed area, the perimeter of a regular polygon with lesser number of sides will always be more than that of a regular polygon with a greater number of sides.
  • If the circumference of a circle is the same as the perimeter of a regular polygon, then the area of the circle will always be more than the area of the polygon.

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